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+COVID-NET USPRO: AN OPEN-SOURCE EXPLAINABLE
+FEW-SHOT DEEP PROTOTYPICAL NETWORK TO MONITOR AND
+DETECT COVID-19 INFECTION FROM POINT-OF-CARE
+ULTRASOUND IMAGES
+Jessy Song
+Department of Systems Design Engineering
+University of Waterloo
+Waterloo, ON N2L 3G1, Canada
+Ashkan Ebadi
+Digital Technologies Research Centre
+National Research Council Canada
+Toronto, ON M5T 3J1, Canada
+[email protected]
+Adrian Florea
+Department of Emergency Medicine
+McGill University
+Montreal, QC H4A 3J1, Canada
+Pengcheng Xi, Stéphane Tremblay
+Digital Technologies Research Centre
+National Research Council Canada
+Ottawa, ON K1A 0R6, Canada
+Alexander Wong
+Department of Systems Design Engineering
+University of Waterloo
+Waterloo, ON N2L 3G1, Canada
+ABSTRACT
+As the Coronavirus Disease 2019 (COVID-19) continues to impact many aspects of life and the global
+healthcare systems, the adoption of rapid and effective screening methods to prevent further spread of
+the virus and lessen the burden on healthcare providers is a necessity. As a cheap and widely accessible
+medical image modality, point-of-care ultrasound (POCUS) imaging allows radiologists to identify
+symptoms and assess severity through visual inspection of the chest ultrasound images. Combined
+with the recent advancements in computer science, applications of deep learning techniques in
+medical image analysis have shown promising results, demonstrating that artificial intelligence-based
+solutions can accelerate the diagnosis of COVID-19 and lower the burden on healthcare professionals.
+However, the lack of a huge amount of well-annotated data poses a challenge in building effective
+deep neural networks in the case of novel diseases and pandemics. Motivated by this, we present
+COVID-Net USPro, an explainable few-shot deep prototypical network, that monitors and detects
+COVID-19 positive cases with high precision and recall from minimal ultrasound images. COVID-
+Net USPro achieves 99.65% overall accuracy, 99.7% recall and 99.67% precision for COVID-19
+positive cases when trained with only 5 shots. The analytic pipeline and results were verified by our
+contributing clinician with extensive experience in POCUS interpretation, ensuring that the network
+makes decisions based on actual patterns.
+Keywords Ultrasonic imaging · Lung · COVID-19 · Few-shot learning · Deep explainable architecture
+1
+Introduction
+The Coronavirus Disease 2019, or COVID-19, caused by severe acute respiratory syndrome coronavirus 2 (SARS-
+CoV-2), has been continuously impacting individual’s well-being and the global healthcare systems [1]. Despite the
+arXiv:2301.01679v1  [eess.IV]  4 Jan 2023
+
+Song et al. (2023). COVID-Net USPro.
+vaccination efforts, policies and regulations in place, due to the rapid transmission of the virus and waves of rising cases,
+the development of effective screening and risk stratification methods remains to be a critical need in controlling the
+disease [2]. Various types of diagnostic tools, including reverse transcription-polymerase chain reaction (RT-PCR),
+rapid antigen detection tests, and antibody tests, have been developed and adapted globally to increase the rate of
+screening. While RT-PCR has been the gold standard test for diagnosing COVID-19, the technique involves large
+labour and laboratory resources and is time-consuming [3]. Other rapid antigen tests and antibody tests with varying
+sensitivity are also less reliable in comparison to RT-PCR tests [3].
+For people with significant respiratory symptoms, medical imaging is used to identity the disease and assess the
+severity of the disease progression [4]. Under this protocol, a computed tomography (CT) scan, chest X-ray (CXR), or
+point-of-care ultrasound (POCUS) imaging can be performed and used clinically as an alternative diagnostic tool [2]. To
+make a diagnosis, acute care physicians and radiologists visually inspect the radiographic images to find patterns related
+to symptoms and to assess severity of COVID-19 infection and deformation [3]. During times of high transmission rate
+of COVID-19, large influx of patients increases the burden on clinicians and radiologists. Medical image processing and
+artificial intelligence (AI) can assist in reducing this burden and accelerate the diagnostic and decision-making process,
+as existing models and algorithms continue to improve and the amount of available medical image data continues to
+grow [5, 6, 7].
+Different imaging modalities, including CT scan, X-ray, and ultrasound may be used in the diagnosis of COVID-19 and
+offer varying diagnostic values [8]. Chest CT scan is the most sensitive imaging modality in the initial diagnosis and
+management of confirmed cases, but it is more expensive and time-consuming [8, 5]. In contrast, ultrasound imaging is
+more accessible and portable, cheap, and safer as radiation is not involved during the examination, which are desirable
+properties for its usage [8], especially in resource-limited settings/environments/areas/regions.
+Deep learning usually requires a large set of training examples [9, 7, 4]. However, due to the nature of novel diseases,
+the availability of such a huge amount of well-annotated data poses a great challenge to the learning algorithms.
+Few-shot learning is an approach where model is trained to classify new data based on a limited number of samples
+exposed in training [10]. This resembles how humans learn, as we can recognize new object classes from very few
+instances, different from other current machine learning techniques that require large amount of data to achieve similar
+performance [10]. Since the few-shot model requires less data to train, the computational costs in the process is also
+significantly reduced [10]. These properties make it an appropriate and promising approach for COVID-19 or rare
+disease diagnosis. One approach for few-shot learning is metric-based learning. As a few-shot metric-based learning
+approach, prototypical networks (PN) perform classification by computing distances to prototype representations of
+each class [10]. PN has shown state-of-the-art (SOTA) results on other datasets/domains (e.g., [11, 12, 13]), proving that
+some simple design decisions can yield significant improvements over other complicated architectures and meta-learning
+approaches [10].
+Motivated by the needs for fast and effective alternative screening solutions and considering ultrasound imaging
+advantages, we present an open-source explainable deep prototypical network, called COVID-Net USPro, that learns to
+detect COVID-19 positive cases with high precision and recall from a very limited number of lung ultrasound (LUS)
+images. When trained with only 5 shots, COVID-Net USPro classifies between positive and negative COVID-19
+cases with 99.65% overall accuracy, 99.7% recall and 99.67% precision for COVID-19 positive cases. Intensive
+experimentation was conducted (e.g., testing different image encoders, varying training conditions and number of
+classes to optimize the network) to assess the performance of COVID-Net USPro network. To ensure the network’s
+fairness and accountability, network benefits from an explainability module, assessing decisions with visual explanation
+tools, i.e., Grad-CAM [14] and GSInquire [15]. Moreover, our contributing clinician (A.F.) carefully verified and
+validated the pipeline and produced results to ensure the validity of the proposed solution from the clinical perspective.
+1.1
+Related Work
+There are several studies that aim to apply deep learning into the screening and detection of COVID-19 positive cases.
+As an open-source and open-access initiative, the COVID-Net [16, 5, 9, 7] includes research on the application of deep
+learning neural networks using multitude of image modalities, such as CT, X-ray, and ultrasound images. Multiple
+works have demonstrated the effectiveness of deep learning in the classification of CT and X-ray images. For example,
+COVID-Net CXR [17], a tailored deep convolutional neural network (DCNN/CNN) for detection of COVID-19 cases
+from chest X-ray images, has achieved an overall accuracy of 98.3% and 97.5% sensitivity for COVID-19 cases.
+Another work by Ozturk et al. proposed a DCNN based on the DarkNet model used for the you only look once (YOLO)
+real time object detection system to classify X-ray images, which achieves 98.08% accuracy for binary COVID-19
+cases detection [18]. Research by Afshar et al. proposed a capsule CNN-based network called COVID-CAPS [19]
+which achieved over 98% accuracy and specificity using a limited amount of X-ray images. COVID-Net CT [6], a
+deep neural network for detection of COVID-19 from CT images, scored 96.2% in sensitivity and 99% in specificity
+2
+
+Song et al. (2023). COVID-Net USPro.
+for COVID-19 cases. Potential of including both CT-scan and X-ray images for classification is also explored, with
+research by Thakur and Kumar demonstrating a DCNN-based model achieving over 99% accuracy and precision for
+COVID-19 detection using images of both modalities [20]. For ultrasound images, custom neural network such as
+COVID-Net US [7] was constructed and tailored to COVID-19 case detection. The network achieved an area under
+receiver operating curve (AUC) of over 98% when trained with positive COVID-19 and negative normal case images.
+Research by Diaz-Escobar et al. [21] also leveraged pre-trained neural networks such as VGG19 [22], InceptionV3
+[23], and ResNet50 [24] in the detection of COVID-19 using ultrasound images and achieved 89.1% accuracy and AUC
+of 97.1%. One limitation of using a custom deep neural network in most of the existing research is the need for a large
+amount of training data, where in mentioned works above, datasets all surpassed 10,000 total images [7, 9].
+Application of few-shot learning techniques has also been investigated. For example, MetaCOVID, proposed by
+Shorfuzzaman et al [25], is a Siamese neural network framework with contrastive loss for few-shot diagnosis of
+COVID-19 infection using CXR images. The performance of the best network achieved an accuracy of 95.6% and AUC
+of 97% when trained under a 3-way, and tested in a 10-shot setting [25]. In [26], a deep siamese convolutional network,
+called COVID-Net FewSE, is able to detect COVID-19 positive cases with 90% recall and accuracy of 99.7% when the
+network is provided with only 50 observations in the training phase. In the work by Karnes et al. [27], the possibility
+of using adaptive few-shot learning for ultrasound COVID-19 detection is examined, and the increasing performance
+with the increasing number of shots is investigated. Although the feasibility of adopting few-shot learning techniques
+for COVID-19 detection from medical imaging has been already investigated, analysis on network’s interpretability
+is either missing or inadequate and lacks clinician validation, which limits the full understanding of the network and
+whether data interpretation process aligns with real clinical settings.
+Our contribution is at least three folds: 1) We presents a high-performing network (99.65% accuracy) trained with only
+5 shots, while other works achieving similar performance require larger numbers of training examples, 2) COVID-Net
+USPro is an explainable network, as demonstrated by analysis from two explainability visualization tools and clinician
+validation, and 3) COVID-Net USPro is open-sourced and available to the public, which helps promote reproducibility
+and accessibility of AI in healthcare and encourage further innovation.
+The remainder of this paper is as follows. Section 2 explains data, techniques, and the experiments conducted to assess
+the network performance in details. Section 3 presents findings from the analysis. Findings are then discussed in
+Section 4 where some limitations of the research and future directions are also presented.
+2
+Data and Methodology
+2.1
+Data
+The COVIDx-US dataset v1.4. [1] is used for this study. COVIDx-US is an open-access benchmark dataset of lung
+ultrasound imaging data that contains 242 videos and 29,651 processed images of patients with COVID-19 infection,
+non-COVID-19 infection, other lung conditions, and normal control cases. The dataset provides LUS images captured
+with two kinds of probe, linear probe which produces a square or rectangular image, or convex probe, which allows for
+a wider field of view [28]. Due to the difference in field of view and low numbers of COVID-19 positive examples with
+linear probe, combining the linear and convex probe data in training may increase noise and influence the performance
+of the network and hence, linear probe data are excluded in this study. A total number of 25,262 convex LUS images are
+then randomly split into train set containing 90% of images in each class and test set with the remaining 10% of images,
+ensuring all frames from each video are either in train or test set to avoid data leakage. All images are rescaled to
+224 × 224 pixels to keep the images across entire dataset consistent. The dataset is further augmented by rotating each
+image by 90°, 180°, 270°, resulting in a total of 101,048 images (25262 × 4). This rotation technique is an appropriate
+method for increasing the dataset size, as it keeps the images and areas of interest for clinical decisions unaltered and
+in-bound [29].
+2.2
+Methodology
+COVID-Net USPro is a prototypical few-shot learning network that trains in an episodic learning setting, using a
+distance metric for assessing similarities between a set of unlabelled data, i.e., query set, and labelled data, i.e., support
+set. Labelled data can be used to compute a single prototype representation of the class, and unlabelled data are assigned
+to the class of the prototype they are closest to. A prototypical network [10] is based on this idea that there exists
+an embedding in which points in a class cluster around a single prototype representation for the class. During the
+training phase, a neural network is used to learn the non-linear mapping of the inputs to an embedding space, and a
+class prototype is computed as the mean of its support set data in the embedding space. Classification is then done by
+finding the nearest class prototype for each query point based on a specified distance metric. An episodic approach
+3
+
+Song et al. (2023). COVID-Net USPro.
+Figure 1: High-level conceptual flow of the Analysis.
+is used to train the model, where in each training episode, the few-shot task is simulated by sampling the data point
+in mini-batches to make the training process consistent with the testing environment. Performance of the network is
+evaluated using the test dataset, and both quantitative analysis based on accuracy, precision and recall and qualitative
+explainability analysis are conducted. An high-level conceptual flow of the Analysis is presented in Figure 1.
+We defined the classification problem as a K-way N-shot episodic task, where K denotes the number of classes present
+in the dataset and N denotes the number of available image examples for each class in each episode. For a given dataset,
+N images from each of the K classes are sampled to form the support set, and another M images from each class are
+sampled to form the query set. The network then aims to classify the images of the query set based on the K ∗ N total
+images presented in the support set. In this work, we formulated the problem as a 2-way, 3-way and 4-way classification
+problem. Details are included under section 2.3.3.
+The few-shot classification with prototypical network can be summarized into three steps: 1) encoding of the images, 2)
+generating class prototypes, and 3) assigning labels to query samples based on distance to the class prototypes. Let’s
+S = {(x(1,s), y(1,s)), . . . , (x(N,s), y(N,s))} and Q = {(x(1,q), y(1,q)), . . . , (x(N,q), y(N,q))} be the support and query
+sets respectively, where each xi ∈ RD is a D-dimensional example feature vector and yi ∈ {1, . . . K} is the label of
+the example. The prototypical network embodies an image encoder fφ : RD → RH that transforms each image xi
+onto a H-dimensional embedding space where images of the same class cluster together. Class prototypes are then
+generated for each class by averaging the embedding image vectors in the support set, where vk = 1
+N
+�N
+i=1 fφ(xi,s(k))
+denotes the prototype of class k [10]. To classify the query image, a distance metric is used where distances between
+the embedding vector of a query image and each of the class prototypes are computed. In this work, squared Euclidean
+distance d (v, q) = ||v − q|| =
+�� (vi − q)2 is used, where q is the embedding vector of the query image and vi is the
+embedding vector of the i-th prototype. After distances are computed, a SoftMax function is applied over distances to
+the prototypes to compute the probabilities of the query image being in each class. The class with the highest probability
+is then assigned to the query image.
+In the training phase, the network learns by minimizing a loss function, i.e., the negative log-SoftMax function
+(J = − log (p (y = k|x))) of the true class k via an optimizer for which we use an Adam optimizer with an initial
+learning rate of 0.001, and reduced if loss is not improved after 3 epochs. In each episode, a subset of data points
+is randomly selected, forming support and query set. Loss term is calculated at the end of each training episode. To
+facilitate effective training process and prevent over-fitting, early stopping is implemented to stop the training process
+4
+
+1. Data Source
+4. Model Construction
+COVIDxUS v1.4 Dataset
+Model Training
+4 classes: COVID, Normal,
+Generate batches of data, build few-shot prototypical network*
+Pneumonia, Other
+and train to perform classification
+Total: 29,651 processed images
+Experiments
+2. Data Preparation
+Adjust 1) image encoder network, 2) training shot settings and 3)
+classification problem formulation to optimize network performance
+Data Selection
+Keep convex probe data
+Total: 25,262 images
+5. Model Evaluation
+Image Preprocessing
+Rescale to 224 × 224 pixels
+Quantitative Evaluation
+Augmentation by rotation of 90°, 180°, 270°
+Evaluate each model's performance with accuracy, precision
+Total: 25,262x4 images
+and recall using the unseen test set
+Select Best-performing model
+10%
+90%
+3. Data Splitting
+Qualitative Explainability Evaluation
+Test
+Train
+Assess model explainability through visual
+COVID: 860 images
+COVID: 7,687 images
+explanation tools
+Normal: 204 images
+Normal: 1,907 images
+Validate results by clinician to ensure diagnosis
+Other: 825 images
+Other: 7,397images
+aligns with clinical perspective
+Pneumonia: 650 images
+Pneumonia: 5,753 imagesSong et al. (2023). COVID-Net USPro.
+Figure 2: COVID-Net USPro, network architecture design.
+when loss term is not improved after 5 epochs. A total of 10 epochs is set for all training processes and 200 episodes is
+set for each training epoch. Figure 2 presents an architecture design overview of the COVID-Net USPro network.
+Trained model’s performance is evaluated quantitatively and qualitatively. In quantitative analysis, model’s accuracy,
+precision and recall for each class are reported. In qualitative analysis, model explainability is investigated and
+visualized. Explainable Artificial Intelligence (XAI) has been an important criterion when assessing whether neural
+networks can be applied to real clinical settings [30]. While AI-driven systems may show high accuracy and precision
+in analyzing medical images, lack of reasonable explainability will spark criticism to the network’s adoption [30].
+COVID-Net USPro’s explainability is assessed using two approached, i.e., Gradient-weighted Class Activation Map
+(Grad-CAM) [14] and GSInquire [15], on a selected dataset containing correctly classified COVID-19 and normal cases
+with high confidence (i.e., > 99.9% probability) as well as falsely predicted COVID-19 and normal cases. Grad-CAM
+generates a visual explanation of the input image using the gradient information flowing into the last convolutional
+layer of the convolutional neural network (CNN) encoder and assigns importance values to each neuron for making a
+classification decision [14]. The output is a heatmap-overlayed image that shows the regions that impact the particular
+classification decision made by the network [14]. The other tool GSInquire identifies the critical factors in an input
+image that are shown to be integral to the decisions made by the network in a generative synthesis approach [15].
+The result is an annotated image highlighting the critical region, which drastically changes the classification result if
+removed [15]. Results from both tools are reviewed by a clinician with experience in analysis of ultrasound images to
+assess whether clinically important patterns are captured by the network.
+2.3
+Experiment Settings
+We comprehensively assess the performance of COVID-Net USPro in detecting COVID-19 positive cases from
+ultrasound images by testing various training conditions such as image encoders, number of shots available for training,
+and classification task types. Details are further discussed in this section.
+2.3.1
+Image Encoders
+To leverage the power of transfer learning, multiple encoders are experimented, including but not limited to the ResNet
+and VGG-based models [24, 22]. Pre-trained models refer to using model parameters pre-trained on ImageNet [31].
+Here, we report 4 best encoders with respect to our research objectives:
+• ResNet18L1: Pre-trained ResNet18 [24], with trainable parameters on the final connected layer and setting
+out features as the number of classes. Model 1 is regarded as the baseline model for encoders, as it contains
+the least number of layers and retrained parameters.
+• ResNet18L5: Pre-trained ResNet18 [24], with trainable parameters on the last 4 convolutional layers and final
+connected layer. Out features set to the number of classes.
+• ResNet50L1: Pre-trained ResNet50 [24], with trainable parameters on the final connected layer and setting
+out features as the number of classes.
+• ResNet50L4: Pre-trained ResNet50 [24], with trainable parameters on the last 3 convolutional layers and final
+connected layer. Out features set to the number of classes.
+5
+
+Prototype Generation &
+Distance Calculation
+Embedding
+Support Set
+Predictions
+Encoder
+In training:
+Loss Calculation
+Query Set
+BackpropagationSong et al. (2023). COVID-Net USPro.
+2.3.2
+Number of Training Shots
+The optimal number of shots for maximized performance is tested by training models under 5, 10, 20, 30, 40, 50, 75,
+and 100-shot scenarios. For selected models showing steady increase of performance over increasing shots, 150 and
+200-shot conditions are tested to verify that the maximum performance is reached at 100-shot. To ensure training
+process is faithful to the testing environment, the number of example shots for each class presented in each episode is
+the same in support and query set in both training and testing. For example, in 5-shot scenario, 5 images in each class
+are presented for both support set and query set in training, and the same follows in testing.
+2.3.3
+Problem Formulation
+As the ability of the model to correctly identify COVID-19 positive cases is valued the most in comparison to other
+classes, the classification problem for identifying COVID-19 was formulated in 3 different scenarios as follows, in an
+ascending order of data complexity:
+• 2-way classification: Data from all 3 other classes, namely ’normal’ class, ’non-COVID-19’ class and ’other’
+class, are viewed as a combined COVID-19 negative class. The network learns from COVID-19 positive and
+COVID-19 negative dataset in this setting.
+• 3-way classification: As the ’other’ class contains data from multiple different lung conditions, it has the
+highest variations and may disrupt network’s learning process due to the lack of uniformity in the data
+compared with COVID-19, normal or non-COVID-19 class. In 3-class classification, the ‘other’ class is
+excluded, and the network is trained to classify the remaining three classes.
+• 4-way classification: As the dataset contains four classes, the four-class classification condition remains this
+setting and network is trained to classify ’COVID-19’, ’normal’, ’non-COVID-19’ and ’other’ class.
+3
+Results
+This section summarizes the quantitative performance results of all combination of experiment settings listed in Section
+2.3 as well as the results of the network explainability analysis.
+3.1
+Quantitative Performance Analysis
+The performance of COVID-Net USPro is evaluated using the overall accuracy, and the precision and recall for each
+class. As the performance of the model to diagnose COVID-19 positive cases is the most important for current clinical
+use case, precision and recall for only COVID-19 case is reported below. To reduce table size, Table 1 only summarizes
+the performance of the network under 5-shot and 100-shot scenarios for encoders that scored over 80% across all
+evaluated metrics. For full performance results of all shot settings and precision, recall for all classes, please refer to
+project repository: [www.anonymous].
+Across all classification types and models, performance is better under 100-shots training scenario than in 5-shot, with
+performance metrics increasing from 5-shot and plateauing after 75-shot, as shown in Figure 3. ResNet networks
+demonstrate the ability to classify COVID-19 with precision and recall above 87% consistently under both 5-shot and
+above 99% under 100-shot condition. As seen in Table 1, the increasing classes in 3-way and 4-way classification
+types reduces the performance of the network, as the classification is more complex given larger number of classes.
+However, this performance difference among the three classification types is reduced when the number of shots
+increases, as more examples available in training improves the network’s ability to distinguish between multiple classes.
+Among the four models, deeper models (i.e., those with ResNet50 as encoder) perform better in all classification types
+and shot conditions. In addition, models with re-trained final convolutional layers parameters (model ResNet18L5
+and ResNet50L4) using the ultrasound images achieve higher accuracy, precision, and recall. Therefore, it can be
+said that while using pre-trained parameters and simpler models reduce the computational complexity and space,
+tailoring parameters on the final 3-4 convolutional layers to the ultrasound images and deeper image encoding boosted
+performance to above 99%.
+In 2-way and 3-way classification, it is also observed that the precision and recall for classes other than COVID-19
+achieve similar magnitude as the COVID-19 class. In the 4-way case, the precision and recall for ‘other’ class is
+around 2-3% lower than those for ‘non-COVID-19’, ‘normal’ and ‘COVID-19’ classes. This is expected since the
+‘other’ class covers various lung conditions/diseases that encompass a larger range of image features and variations.
+Overall, with precision and recall achieving similar magnitude for all cases in 2-way, 3-way and 4-way classification,
+the network also demonstrates the ability to distinguish between multiple diseases. In comparison to studies outlined in
+6
+
+Song et al. (2023). COVID-Net USPro.
+Table 1: Summary of classification results for 5-shot and 100-shot conditions.
+Scenario
+No. shots
+Model
+Accuracy
+Precision
+Recall
+2-way
+5
+ResNet18L1
+0.9420
+0.9486
+0.9460
+2-way
+5
+ResNet18L5
+0.9930
+0.9925
+0.9950
+2-way
+5
+ResNet50L1
+0.9525
+0.9570
+0.9560
+2-way
+5
+ResNet50L4
+0.9965
+0.9967
+0.9970
+2-way
+100
+ResNet18L1
+0.9758
+0.9764
+0.9755
+2-way
+100
+ResNet18L5
+1.0000
+1.0000
+1.0000
+2-way
+100
+ResNet50L1
+0.9963
+0.9964
+0.9962
+2-way
+100
+ResNet50L4
+0.9999
+0.9999
+1.0000
+3-way
+5
+ResNet18L1
+0.9570
+0.9606
+0.9510
+3-way
+5
+ResNet18L5
+0.9987
+0.9992
+0.9970
+3-way
+5
+ResNet50L1
+0.9945
+0.9508
+0.9660
+3-way
+5
+ResNet50L4
+0.9947
+0.9942
+0.9940
+3-way
+100
+ResNet18L1
+0.9867
+0.9833
+0.9853
+3-way
+100
+ResNet18L5
+1.0000
+1.0000
+1.0000
+3-way
+100
+ResNet50L1
+0.9977
+0.9970
+0.9975
+3-way
+100
+ResNet50L4
+1.0000
+1.0000
+1.0000
+4-way
+5
+ResNet18L1
+0.8627
+0.9281
+0.8710
+4-way
+5
+ResNet18L5
+0.9817
+0.9975
+0.9970
+4-way
+5
+ResNet50L1
+0.9392
+0.9640
+0.9540
+4-way
+5
+ResNet50L4
+0.9850
+0.9917
+0.9930
+4-way
+100
+ResNet18L1
+0.9385
+0.9742
+0.9704
+4-way
+100
+ResNet18L5
+0.9884
+1.0000
+1.0000
+4-way
+100
+ResNet50L1
+0.9813
+0.9947
+0.9955
+4-way
+100
+ResNet50L4
+0.9902
+1.0000
+1.0000
+Figure 3: Performance results with increasing shots trained under 4-class condition: (a) Pre-trained ResNet18 with
+trainable parameters on the final connected layer and setting out features as the number of classes (ResNet18L1).
+(b) Pre-trained ResNet50 with trainable parameters on the last 3 convolutional layers and final connected layer
+(ResNet50L4).
+7
+
+1.00
+1.01
+0.98
+1.00
+0.96
+0.99
+0.94
+0.98
+0.92
+Metric
+0.97
+Metric
+0.90
+Accuracy
+Accuracy
+COVID-19 Precision
+0.96
+COVID-19 Precision
+0.88
+COVID-19 Recall
+COViD-19 Recal
+0.86
+0.95
+50
+100
+150
+0
+200
+0
+50
+100
+150
+200
+Shots
+Shots
+(a) ResNet18L1
+(b) ResNet50L4Song et al. (2023). COVID-Net USPro.
+Figure 4: COVID-19 positive case examples correctly classified by COVID-Net USPro with high confidence: (a) an
+example of wrong decision factors. (b) an example of a decision made based on the disease-related patterns.
+Section 1.1, the performance of COVID-Net USPro networks tailored to ultrasound images with re-trained parameters
+is improved. Accuracy of ResNet50L1 and ResNet50L4 exceeds 98% under 4-way 5-shot setting, while other work
+such as MetaCOVID [25], which also applied a few-shot approach, achieved 95.6% accuracy under 3-way, 10-shot
+setting. Additionally, the sensitivity of COVID-Net USPro for COVID-19 cases are also higher than networks trained
+with other image modality data such as X-ray or CT, where they scored 97.5% in the best performing case [6].
+3.2
+Clinical Validation and Network Explainability Analysis
+In addition to the intensive quantitative performance analysis, we clinically validated the network output to ensure that
+the network captures important patterns in the ultrasound images. For this purpose, our contributing clinician (A.F.)
+reviewed a randomly selected set of images and reported his findings and observations. Our contributing clinician (A.F.)
+is an Assistant Professor in the Department of Emergency Medicine and the ultrasound co-director for undergraduate
+medical students at McGill University. He is practicing Emergency Medicine full-time at Saint Mary’s Hospital in
+Montreal.
+Figure 4 presents two select ultrasound images of COVID-19 positive cases, annotated by Grad-CAM and GSInquire,
+as examples, that were reviewed. As seen, the annotated images contain the lung pleura region at the top of the
+image, while the second example (Figure 4-b) also marks the bottom region with high importance. B-lines, or the light
+comet-tail artifacts extending from pleura to the bottom of the image, and the presence of dark regions interspacing
+the B-lines at the bottom part of the image corresponding to signs of lung consolidation are indicators of abnormality
+[32]. Hence, the visual annotations for the second example (Figure 4-b) are more representative for disease-related
+patterns within the ultrasound image. Figure 4-a is one of the examples where the model considers the rib as a structure
+of interest, which is not the abnormality, leading to classify the images as a COVID-19 positive case. Hence, although
+the model correctly classified the image, the decision was made based on invalid clinical factors.
+We implement two strategies to solve the mentioned issues and improve classification explainability. First, excluding
+images with low image quality, such as insufficient image depth or the lack of representative features. A severity grade
+introduced by COVIDx-US dataset v1.4, called lung ultrasound score (LUSS), rates each ultrasound video on a scale
+of 0 to 3, where 0 corresponds to presence of only normal features, and 3 corresponds to presence of severe disease
+artifacts [33]. Therefore, in the first attempt to improve the network, images from videos with score of 0 for the normal
+case and images from videos with score of 2 and 3 for COVID-19 case are used to train a binary classification version
+8
+
+GSInquire
+Grad-CAM
+Original Image
+Annotated Image
+Annotated Image
+a)
+b)Song et al. (2023). COVID-Net USPro.
+Figure 5: Four cropped COVID-19 positive examples predicted correctly with high confidence by COVID-Net USPro
+(a-d), while recognizing disease artifacts, e.g., extended B-lines.
+of the network. By observing the annotated images, network shows to focus more on the bottom regions of the images,
+though cases where network focus on the top pleura region are still present. The second strategy to further improve
+model explainability is to exclude regions above the pleura (i.e., soft tissue) of the images, so that network focuses on
+the disease-defining features, present mostly at the bottom of the images below lung pleura. Our experiments confirm
+the effectiveness of this strategy. Hence, combining the first and second strategy, a binary model with LUSS score
+filtered and cropped images is trained. Figure 5 shows examples from the cropped images analysis. As suggested from
+the annotated examples and confirmed by our contributing clinician (A.F.), clinically determining artifacts such as
+B-lines and lung consolidation are clearly identified in COVID-19 positive images by COVID-Net USPro.
+9
+
+Grad-CAM
+GSInquire
+Original Image
+Annotated Image
+Annotated Image
+a)
+b)
+c)
+d)Song et al. (2023). COVID-Net USPro.
+4
+Conclusions
+Deep neural network architectures have shown promising results in a wide range of tasks, including predictive and
+diagnostic tasks. However, such networks require a massive amount of labelled data to train which is against the nature
+of new pandemics and novel diseases where there are no or very few data samples available, especially in the initial
+stages. As part of the COVID-Net initiative and using a diverse complex benchmark dataset, i.e., COVIDx-US, in this
+work we introduce the COVID-Net USPro network, tailored to detect COVID-19 infection with high accuracy from very
+few ultrasound images. The proposed deep prototypical network leverages pretrained models with tailored parameters
+on final layers to reduce computational complexity and achieve high classification performance when only 5 examples
+from each class are available for training. Accuracy, precision and recall for the best performing network are over 99%,
+which are comparable or outperforming other existing work [7, 27]. These properties are not only highly crucial for the
+control of the COVID-19 pandemic but also for screening patients in new diseases/pandemics for which the proposed
+network can be easily tuned. We intensively assessed the explainability of the network and clinically validated its
+performance. Experimental results demonstrate that COVID-Net USPro can not only achieve high performance in terms
+of accuracy, precision, and recall, but also shows predictive behaviour that is consistent with clinical interpretation, as
+validated by our contributing clinician (A.F.). In addition, as part of the explainability-driven performance validation
+process, we proposed and implemented two strategies to further improve the network performance in accordance with
+the background clinical knowledge in identifying COVID-19 positive and negative cases. Overall, we believe the
+simplicity and effectiveness of COVID-Net USPro makes it a promising tool to aid the COVID-19 screening process
+using ultrasound images. We hope the open-source release of COVID-Net USPro help researchers and clinical data
+scientists to accelerate innovations in the combat against the COVID-19 pandemic that can ultimately benefit the larger
+society.
+Several future research directions can be explored to further improve the network. First, some additional steps in data
+augmentation and preparation can be taken to improve data quality and dataset size. In this work, ultrasound images
+captured with linear probe are excluded due to differences in clinical interpretation of linear probe and convex probe
+captured images. More image augmentation and preparation techniques can be experimented to include linear probe
+data and increase the data size. Second, in this work, we used simple cropping to filter out the pleura region of the
+images. A more procedural image segmentation step could be added to include only clinically relevant areas of the
+images for network construction to further improve network performance from the explainability standpoint. Lastly, we
+used COVIDx-US which is a public dataset that includes data of various sources and quality. Network training could be
+improved by only using high quality input ultrasound data, collected systematically, which contain clear representative
+image artifacts with sufficient/specific image depth. For this purpose, a data collection protocol might be required to
+capture ultrasound images in a standardized manner from a set of consented participants.
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+12
+

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+arXiv:2301.03733v1  [quant-ph]  10 Jan 2023
+Design Optimization of Noise Filter using Quantum Annealer
+Akihisa Okada,1, ∗ Hiroaki Yoshida,1 Kiyosumi Kidono,1
+Tadayoshi Matsumori,2 Takanori Takeno,2 and Tadashi Kadowaki2
+1TOYOTA CENTRAL R&D LABS., INC., Bunkyo-ku, Tokyo 112–0004, Japan
+2DENSO CORPORATION, Minato-ku, Tokyo 108–0075, Japan
+(Dated: January 11, 2023)
+The use of quantum annealers in black-box optimization to obtain the desired properties of a
+product with a small number of trials has attracted attention. However, the application of this
+technique to engineering design problems is still limited. Here, we demonstrate the applicability of
+black-box optimization with a quantum annealer to the design of electric circuit systems, focusing
+on π-type noise filters as an example. We develop a framework that uses quantum annealing to find
+the optimal location of electrical components and conductor paths connecting the components, and
+confirm that the learning process appropriately works over a number of trials to efficiently search for
+a design with high performance. The results show the potential applicability of quantum annealing
+to design problems of electric circuit systems.
+Keywords: Combinatorial optimization problem, Noise filter, Quadratic unconstrained binary optimization,
+Quantum annealing, Quantum computing
+I.
+INTRODUCTION
+A.
+Quantum annealers
+High-performance computers are required to elucidate
+and predict complex phenomena, such as in simulations
+of the behavior of systems with multiple interconnected
+factors. However, Neumann-type computers, whose de-
+velopment has followed Moore’s law, do not meet the
+demand for high performance. Drastic improvements in
+Neumann-type computers are not expected [1] as their
+single-threaded performance has reached its ceiling [2].
+Therefore, non-Neumann-type computers are expected
+to be an alternative for high-performance computation
+for complex problems.
+Quantum annealers are one type of non-Neumann-type
+computer. Commercial machines are available from D-
+Wave Systems. The architecture of a quantum annealer
+implements the Ising model on a circuit using supercon-
+ductivity. The ground state of the Ising model is effi-
+ciently found using the quantum effect [3].
+Since the
+ground state of the Ising model is equivalent to the so-
+lution of quadratic unconstrained binary optimization
+(QUBO), which includes not only fundamental prob-
+lems [4] but also practical ones [5–10], a quantum an-
+nealer is regarded as a quantum solver for QUBO prob-
+lems.
+Pragmatically, the usability of quantum annealers for
+complex problems relies on their compatibility with the
+QUBO formulation. Previous studies are limited to cases
+in which the original problem formulation has an appar-
+ent link to QUBO, such as that for combinatorial opti-
+mization problems. A recent study combined quantum
+annealing with machine learning to find the optimal ar-
+∗ [email protected]
+rangement of the constituent elements of a metamate-
+rial [11]. The original problem (optical properties of the
+metamaterial) was not necessarily converted to a QUBO
+formulation, implying the applicability of quantum an-
+nealing to general optimization problems.
+Specifically,
+they proposed a type of black-box optimization frame-
+work, in which the unknown relation between the input
+binary variables and the complex property values com-
+puted according to the governing equations is learned
+by means of a second-order regression equation and the
+optimal input variables are obtained using quantum an-
+nealing.
+Reports of applying black-box optimization to design
+problems are limited to optical problems with the opti-
+mal arrangement of metamaterials described above and
+photonic-crystals [12], the structural dynamics problem
+of substrate vibration [13], and molecular design [14].
+B.
+Design problem of noise filter
+In this study, we focus on an electric noise filter as
+an example electric circuit. Noise filter performance de-
+pends on the combination of electrical components and
+the paths of conductors connecting them.
+Products designed for electromagnetic compatibility
+incorporate noise filters that reduce input voltage noise to
+prevent high-frequency noise from affecting surrounding
+electronic devices. The electrical component allocation
+region needs to be determined under the constraint of a
+certain amount of noise attenuation, i.e., an optimal filter
+design is required. In this study, we apply black-box opti-
+mization that incorporates calculations conducted using
+quantum annealing to the design optimization of a noise
+filter that consists of two capacitors and an inductor,
+called a π-type filter, and demonstrate that this opti-
+mization framework is useful for electric circuit design
+problems.
+
+2
+Capacitor 1
+Capacitor 2
+Inductor
+Ground
+Voltage source
+with noise
+Output port
+Input port
+FIG. 1. Circuit diagram of π-type noise filter.
+Topology optimization has been used for optimal de-
+sign.
+Although topology optimization is applicable to
+electric circuits [15], the inherent challenge is to avoid
+falling into a local optimal solution, which stems from the
+method being based on the gradient method. In particu-
+lar, optimization problems with many degrees of freedom
+related to element location, as considered in this study,
+generally have a complex objective function space, which
+can hinder the search for the global optimal solution.
+The proposed optimization framework, which combines
+black-box optimization and quantum annealing, exploits
+the features of quantum annealing to avoid becoming
+trapped in a local optimal solution.
+C.
+Summary of contributions
+The contributions can be summarized as follows.
+• We extend the framework of optimal design based
+on black-box optimization using quantum anneal-
+ing to problems related to electric circuit systems.
+• We confirm that the optimization process works as
+an optimal design method for electric circuits by
+analyzing the learning process based on the relation
+between the number of searches and performance
+values.
+II.
+METHOD
+A.
+Design problem of π-type noise filter
+A circuit diagram of the π-type noise filter to be de-
+signed is shown in Fig. 1. The circuit consists of three
+elements, namely an inductor and two capacitors. Fig-
+ure 2 shows the π-type noise filter model utilized in this
+study. It is assumed that the back side of the substrate is
+grounded. The performance of a noise filter is determined
+by the capacitance of the capacitor, the inductance of the
+inductor, inductive noise, and parasitic capacitance. The
+inductive noise and parasitic capacitance depend on the
+relative location of the inductor, the capacitors, and the
+conductor path, which does not appear in the circuit di-
+agram but should be designed as described below.
+Substrate
+Inductor
+Capacitor 1
+Capacitor 2
+Conductor
+Output port
+Input port
+FIG. 2. Example of element and conductor arrangement for
+π-type noise filter. The input and output ports, capacitors,
+and inductor are represented by simple square elements. The
+backplane is the electrical ground.
+x
+y
+Data acquisition and learning
+Hidden true system
+y = f(x)
+y = xTAx
+(1)
+(2)
+(3)
+~
+FIG. 3. Schematic diagram of BOCS. (1) Data y for input
+x is obtained from simulation or experiment.
+(2) Second-
+order regression equation is estimated from input x and y.
+˜y is estimated value.
+(3) Optimal x is found.
+Here, A is
+the coefficient of the quadratic regression equation. f is an
+unknown function under the governing equation.
+B.
+Black-box optimization of noise filter
+The objective of black-box optimization is to obtain
+the input parameter x that minimizes (or maximizes)
+the characteristic value y with a small number of tri-
+als under the condition that the relation between x and
+y (y = f(x)) is unknown. Here, we focus on Bayesian
+Optimization of Combinatorial Structures (BOCS) [16],
+which is a learning method applicable to cases where the
+input parameter x is a binary variable, as done in the
+literature [17]. In BOCS, the relation between x and y
+is learned sequentially using a quadratic regression equa-
+tion of x.
+In other words, starting with several data
+sets of x and y, we (1) obtain the data y for the input
+x through simulations or experiments on a real system
+where the input-output relation is unknown, (2) learn
+the relation between data y and input x in quadratic
+form, and (3) search for the optimal input x under the
+assumed quadratic relation.
+The relation between the
+various tasks in BOCS is summarized in Fig. 3.
+To apply this black-box optimization to the design of
+noise filters, we define a binary variable x that specifies
+
+3
+Input port
+positions
+Output port
+positions
+Capacitor 1
+positions
+Capacitor 2
+positions
+Inductor
+positions
+A
+B
+C
+X
+Y
+FIG. 4. Candidate element positions and conductor paths.
+As an example of conductor paths, three candidates (A, B,
+and C) between the upper side of the input port and the left
+side of capacitor 1 are shown.
+the location of the element and the conductor path, and
+employ electromagnetic field analysis using the finite el-
+ement method as the data acquisition method in (1). In
+(3), quantum annealing is employed to find the global
+minimum in the regression model, which has many lo-
+cal minima. The solution x of the quantum annealing
+and the corresponding output value y are added to the
+data in the learning process. We refer to this method
+as BOCS-QA. To clarify the effect of quantum anneal-
+ing, a calculation using simulated annealing (BOCS-SA)
+instead of quantum annealing is also performed and the
+results are compared.
+The following sections describe the binary design vari-
+ables that represent electrical component positions and
+conductor paths and the characteristic values for evalu-
+ating filter performance.
+1.
+Binary design variables
+The element positions and conductor paths between
+the elements are mapped to the binary variable x. In
+this study, the problem is to select the positions of five
+elements (an input port, an output port, an inductor, and
+two capacitors) from two candidates and the conductor
+paths from three candidates. In order to represent these
+variables as binary variables, the substrate is divided into
+a 10×15 (X×Y) grid. The input and output ports are
+placed on the sides of the board and the inductor and
+capacitor are placed in the grid as concentrated elements,
+as shown in Fig. 4.
+Three candidate paths as conductors are created by
+connecting the elements from top to bottom in the fol-
+lowing manner.
+A. Draw a path in the X direction and then in the Y
+direction.
+B. Draw a path in the Y direction to half of the dif-
+ference, then in X, and then in the remaining Y
+direction.
+Input port �
+� Output port
+Capacitor 1
+Capacitor 2
+Inductor
+Conductor
+FIG.
+5.
+Circuit
+corresponding
+to
+bit
+string
+“0101101010010001100100” in one-hot representation.
+C. Draw a path in the Y direction and then in the X
+direction.
+The typical π-type noise filter, shown in Fig. 2, is appro-
+priately included as a candidate by the above conductor
+setting. The present method can be simply extended to
+the case with more than three candidate paths if neces-
+sary.
+We adopt one-hot encoding to represent noise filters
+in which element positions and conductor paths are se-
+lected from these candidates. In the case considered here,
+22 bits are required because there are two candidates for
+each of the five element positions and three candidates
+for each of the four conductor paths. Let “10” be the
+state in which the element is at the bottom or on the
+left and “01” be the state in which it is at the top or
+on the right. Then, let “100” be a conductor path that
+first moves in the X direction, “010” be one that turns
+in the middle, and “001” be one that first moves in the
+Y direction. The bits that represent the conductor path
+follow the element position bits; that is, the first 10 bits
+represent the five element positions and the latter 12 bits
+represent the selection of the four conductor paths. The
+bits that represent the element positions are arranged on
+the board in the following order from left to right: in-
+put port, capacitor 1, inductor, capacitor 2, and output
+port. The conductor paths are similarly arranged in the
+following order from left to right: input port - capaci-
+tor 1, capacitor 1 - inductor, inductor - capacitor 2, and
+capacitor 2 - output port.
+For example, a circuit en-
+coded by “0101101010010001100100” as binary variable
+x is shown in Fig. 5.
+2.
+Obtaining characteristic value
+The S-parameter S21 is adopted as the characteristic
+value y of the noise filter. S21 indicates the ratio of out-
+put power to input power. When the input power of noise
+is p1 and the output power is p2, S21 is expressed by the
+following equation,
+S21 =
+�
+|p2|
+|p1|.
+(1)
+
+4
+FIG.
+6.
+Circuit
+corresponding
+to
+bit
+string
+“1001011001000001000100”.
+The
+conductor
+paths
+be-
+tween the input power port and capacitor 1 and those
+between the inductor and capacitor 2 are not selected. To
+avoid disconnection, conductors spread over the board are
+assigned.
+We design a noise filter that minimizes S21 under the
+given noise voltage. p1 and p2 are calculated using finite
+element analysis for simulating the electromagnetic field
+of the electric circuit model shown in Fig. 2, that is, the
+model in which the back of the board is the ground and
+the electrical components are lumped-parameter ones on
+the surface of the board. A sufficiently large air region is
+provided around the board in order to precisely calculate
+the induced noise. A scattering boundary condition is
+set at the outermost boundary of the air region.
+Special procedures are required in the following two
+cases where the S-parameters are not correctly evaluated
+by the finite element method.
+(I) Element position does not satisfy the one-hot con-
+straint.
+(II) A bit in the conductor path is “000” (the circuit
+has a disconnection on the board).
+In case (I), the binary variables are unencodable to a
+configuration of a noise filter. Given such binary vari-
+ables, instead of performing the finite element method,
+we calculate y as a penalty according to the following
+formula,
+y = ybase + λ
+5
+�
+m=1
+(x2m−1 + x2m − 1)2 ,
+(2)
+where ybase is the base value of the violation of one-
+hot constraints, λ is the penalty coefficient, and xi is
+the value of the i-th bit of the binary variable x. Since
+BOCS learns characteristic values in quadratic form, this
+penalty of one-hot constraints is also expected to be
+learned.
+In case (II), a meaningful S-parameters for evaluating
+a noise filter performance cannot be obtained because
+the conductor path is disconnected such that voltage is
+conducted neither from a normal signal nor noise. We
+assign a dummy conductor that avoids the disconnection,
+as shown in Fig. 6. Note that when multiple conductor
+paths are selected, such as “011”, we take the sum of the
+conductor paths.
+To summarize, we calculate the characteristic value of
+a noise filter y using the following equation,
+z ≡
+5
+�
+m=1
+(x2m−1 + x2m − 1)2 ,
+(3)
+y =
+�
+S21
+( for z = 0 ),
+ybase + λz
+( for z ̸= 0 ).
+(4)
+C.
+Parameters for circuit model and black-box
+optimization
+For the calculation of characteristic values, the sub-
+strate thickness, width, and height are set to 1.6, 150,
+and 100 mm, respectively. An air area of 30 mm is pro-
+vided around the board. Scattering boundary conditions
+are set at the outermost boundaries of this air region.
+The substrate is divided into a 10×15 grid, as introduced
+in section II B 1.
+The physical constants of the power supply port, ca-
+pacitor, and inductor are set to 50 Ω, 100 F, and 10 H,
+respectively. The substrate’s relative permittivity, rela-
+tive permeability, and conductivity are set to 4.5, 1, and
+1.0 × 10−8 S/m, respectively, assuming an FR-4 sub-
+strate. The conductor is treated as a perfect conductor.
+In addition, S21 was calculated using a frequency anal-
+ysis at 10 MHz. For Eq. (2), we set ybase = −60 and
+λ = 10.
+The quantum annealer was Advantage system4.1 by
+D-Wave Systems.
+We adopted the Python library
+dwave-neal by D-Wave Systems as a simulated anneal-
+ing method. The sampling number was set to 3000 when
+solving the problem. The x value that gave the small-
+est y was adopted as the next candidate. For the initial
+training datasets of BOCS, we prepared 20 randomly gen-
+erated binary variables x and their corresponding char-
+acteristic values y. BOCS-QA and -SA were performed
+until 300 searches were conducted.
+III.
+RESULTS AND DISCUSSION
+We compare the results of the BOCS-QA and BOCS-
+SA calculations with those of random search in which
+binary variables were randomly generated.
+First, the results of all search histories of BOCS-QA
+and random search are shown in Figs. 7 and 8, respec-
+tively.
+The learning processes of BOCS-QA and random
+search are different.
+Figure 7 shows that BOCS-QA
+mainly learned the penalty term in Eq. (2) in the be-
+ginning (before approximately 60th search), and sub-
+sequently started to learn on the bases of the perfor-
+mance of the noise filter S21, suggesting that the design
+of the penalty term facilitated learning. Then, the high-
+est record of S21 was steadily set. On the other hand,
+
+5
+-120
+-100
+-80
+-60
+-40
+-20
+0
+0
+50
+100
+150
+200
+250
+300
+y
+Number of searches
+FIG. 7. Full search history of BOCS-QA.
+-120
+-100
+-80
+-60
+-40
+-20
+0
+0
+50
+100
+150
+200
+250
+300
+y
+Number of searches
+FIG. 8. Full search history of random search.
+the random search shown in Fig. 8 searched for a feasi-
+ble noise filter in very rare cases. There is no particular
+trend. The number of solutions that satisfy the one-hot
+constraint is ten, which is close to the expected value.
+The probability that a random binary variable satisfies
+the one-hot constraint is 25/210 = 1/32, so the expected
+number for 300 searches is nine.
+The update records of the characteristic value y ver-
+sus the number of searches are shown in Fig. 9. Since
+BOCS-QA, BOCS-SA, and random search are random-
+ized algorithms, the mean, minimum, and maximum val-
+ues were calculated for ten trials. BOCS-QA and BOCS-
+SA steadily search for a noise filter with good perfor-
+mance, whereas random search tends to have a large
+variance (especially with a small number of searches).
+The steady performance improvement of BOCS-QA and
+BOCS-SA shown in Fig. 9 is due to the successful learn-
+ing of S21, as confirmed in Fig.
+7.
+At 300 searches,
+BOCS-QA shows slightly better performance than that
+of BOCS-SA in terms of the average, minimum, and max-
+imum values, as shown in Table I.
+Next, we evaluate the filter performance of the ob-
+tained solution. Since there are 222 cases (expressed in
+22 bits), enumerating the performance of all solutions is
+unrealistic. We therefore choose only the relevant solu-
+tions with unique element positions and a single conduc-
+tor path between elements. This gives a total of 2592
+cases (25 = 32 combinations of element positions and
+34 = 81 combinations of conductor positions).
+Figure
+Best record of y
+Number of searches
+search
+FIG. 9.
+Updated records of y. The solid and dotted lines
+represent the mean and the filled area represents the area
+between the maximum and minimum values.
+0
+50
+100
+150
+200
+250
+300
+350
+400
+450
+[–82, –84)
+[–84, –86)
+[–86, –88)
+[–88, –90)
+[–90, –92)
+[–92, –94)
+[–94, –96)
+[–96, –98)
+[–98, –100)
+[–100, –102)
+[–102, –104)
+[–104, –106)
+[–106, –108)
+[–108, –110)
+[–110, –112)
+Frequency
+S 21 (dB)
+FIG. 10. Histogram of S21 value in decibels when element
+positions are specified uniquely and there is one conductor
+between elements.
+10 shows a histogram of the S21 value in decibels. For
+our settings, noise filters whose S21 is under −108dB are
+rare (approximately 3%). Since the average records of
+BOCS-QA and BOCS-SA are in the top 0.8% and 1.9%,
+respectively, as shown in Table I, these methods finding
+such filters in 300 searches are considered efficient.
+The configuration of the best-performing noise filter
+obtained using BOCS-QA is shown in Fig. 11. In this
+TABLE I. Comparison of results obtained by various meth-
+ods.
+Method
+Object
+Value
+Rank
+QA
+Best
+−111.34 dB
+1st
+Average −109.64 dB
+19th
+Worst
+−106.97 dB 192nd
+SA
+Best
+−110.55 dB
+14th
+Average −108.91 dB
+48th
+Worst
+−104.80 dB 528th
+Random
+Best
+−107.12 dB 180th
+Average −104.80 dB 192th
+Worst
+−102.00 dB 1058th
+
+6
+FIG. 11. Noise filter obtained by BOCS-QA.
+case, the value of S21 was −111.34 dB. The input port
+and capacitor are placed close to each other, prevent-
+ing performance degradation due to induced noise. This
+shows that the obtained configuration is physically rea-
+sonable.
+In this study, we formulated a problem with two candi-
+dates for the element positions and three candidates for
+the conductor paths. If we considered a large-scale prob-
+lem with a larger number of candidates, the probability
+of finding a well-posed noise filter by chance using ran-
+dom search would be much smaller and the superiority
+of BOCS-QA and BOCS-SA would be more significant.
+IV.
+CONCLUSION AND OUTLOOK
+To find input parameters that provide the desired char-
+acteristics with a small number of searches, we proposed
+an iterative optimization method that incorporates quan-
+tum annealing in the BOCS framework and applied it to
+the problem of designing noise filters. A π-type noise fil-
+ter that consists of two capacitors and an inductor was
+considered. A model was created to select two candidates
+for the location of these elements and three candidates
+for the path of the conductor connecting the elements.
+The results show that a high-performance noise filter
+can be efficiently found and that the search progresses
+more stably than does random search. This shows that
+the framework that incorporates quantum annealing into
+black-box optimization is applicable to electric circuit de-
+sign problems. The present method could help engineers
+meet the high demand for electrical products.
+Beyond
+the
+optimization
+of
+electric
+components
+demonstrated here, system-level optimization of electric
+devices is a topic for future work. It could lead to mul-
+tiphysics optimal design that requires simultaneous opti-
+mizations of multiple phenomena.
+The proposed BOCS framework was proven to work
+with quantum annealing and simulated annealing.
+A
+comparison of these two versions showed only a slight
+difference. A recent study that compared the two solvers
+in an black-box optimization framework also concluded
+that clear performance improvements using quantum an-
+nealing are rare [13]. However, a clear advantage of quan-
+tum annealing in finding optimal solutions, achieved by
+adjusting the annealing schedule, has recently been re-
+ported [18]. Future research should thus examine in de-
+tail the scheduling protocols to further improve the per-
+formance of BOCS with quantum annealing. In addition,
+a recent improvement of the learning process [17] could
+be integrated into the present BOCS framework to speed
+up the whole optimization process.
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='03733v1  [quant-ph]  10 Jan 2023  Design Optimization of Noise Filter using Quantum Annealer  Akihisa Okada,1, ∗ Hiroaki Yoshida,1 Kiyosumi Kidono,1  Tadayoshi Matsumori,2 Takanori Takeno,2 and Tadashi Kadowaki2  1TOYOTA CENTRAL R&D LABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=', INC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=', Bunkyo-ku, Tokyo 112–0004, Japan  2DENSO CORPORATION, Minato-ku, Tokyo 108–0075, Japan  (Dated: January 11, 2023)  The use of quantum annealers in black-box optimization to obtain the desired properties of a  product with a small number of trials has attracted attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' However, the application of this  technique to engineering design problems is still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Here, we demonstrate the applicability of  black-box optimization with a quantum annealer to the design of electric circuit systems, focusing  on π-type noise filters as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' We develop a framework that uses quantum annealing to find  the optimal location of electrical components and conductor paths connecting the components, and  confirm that the learning process appropriately works over a number of trials to efficiently search for  a design with high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The results show the potential applicability of quantum annealing  to design problems of electric circuit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Keywords: Combinatorial optimization problem, Noise filter, Quadratic unconstrained binary optimization,  Quantum annealing, Quantum computing  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  INTRODUCTION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Quantum annealers  High-performance computers are required to elucidate  and predict complex phenomena, such as in simulations  of the behavior of systems with multiple interconnected  factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' However, Neumann-type computers, whose de-  velopment has followed Moore’s law, do not meet the  demand for high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Drastic improvements in  Neumann-type computers are not expected [1] as their  single-threaded performance has reached its ceiling [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Therefore, non-Neumann-type computers are expected  to be an alternative for high-performance computation  for complex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Quantum annealers are one type of non-Neumann-type  computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Commercial machines are available from D-  Wave Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The architecture of a quantum annealer  implements the Ising model on a circuit using supercon-  ductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The ground state of the Ising model is effi-  ciently found using the quantum effect [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Since the  ground state of the Ising model is equivalent to the so-  lution of quadratic unconstrained binary optimization  (QUBO), which includes not only fundamental prob-  lems [4] but also practical ones [5–10], a quantum an-  nealer is regarded as a quantum solver for QUBO prob-  lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Pragmatically, the usability of quantum annealers for  complex problems relies on their compatibility with the  QUBO formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Previous studies are limited to cases  in which the original problem formulation has an appar-  ent link to QUBO, such as that for combinatorial opti-  mization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' A recent study combined quantum  annealing with machine learning to find the optimal ar-  ∗ a-okada@mosk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='tytlabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='jp  rangement of the constituent elements of a metamate-  rial [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The original problem (optical properties of the  metamaterial) was not necessarily converted to a QUBO  formulation, implying the applicability of quantum an-  nealing to general optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Specifically,  they proposed a type of black-box optimization frame-  work, in which the unknown relation between the input  binary variables and the complex property values com-  puted according to the governing equations is learned  by means of a second-order regression equation and the  optimal input variables are obtained using quantum an-  nealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Reports of applying black-box optimization to design  problems are limited to optical problems with the opti-  mal arrangement of metamaterials described above and  photonic-crystals [12], the structural dynamics problem  of substrate vibration [13], and molecular design [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Design problem of noise filter  In this study, we focus on an electric noise filter as  an example electric circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Noise filter performance de-  pends on the combination of electrical components and  the paths of conductors connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Products designed for electromagnetic compatibility  incorporate noise filters that reduce input voltage noise to  prevent high-frequency noise from affecting surrounding  electronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The electrical component allocation  region needs to be determined under the constraint of a  certain amount of noise attenuation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=', an optimal filter  design is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In this study, we apply black-box opti-  mization that incorporates calculations conducted using  quantum annealing to the design optimization of a noise  filter that consists of two capacitors and an inductor,  called a π-type filter, and demonstrate that this opti-  mization framework is useful for electric circuit design  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  2  Capacitor 1  Capacitor 2  Inductor  Ground  Voltage source  with noise  Output port  Input port  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Circuit diagram of π-type noise filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Topology optimization has been used for optimal de-  sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Although topology optimization is applicable to  electric circuits [15], the inherent challenge is to avoid  falling into a local optimal solution, which stems from the  method being based on the gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In particu-  lar, optimization problems with many degrees of freedom  related to element location, as considered in this study,  generally have a complex objective function space, which  can hinder the search for the global optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The proposed optimization framework, which combines  black-box optimization and quantum annealing, exploits  the features of quantum annealing to avoid becoming  trapped in a local optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Summary of contributions  The contributions can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  We extend the framework of optimal design based  on black-box optimization using quantum anneal-  ing to problems related to electric circuit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  We confirm that the optimization process works as  an optimal design method for electric circuits by  analyzing the learning process based on the relation  between the number of searches and performance  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  METHOD  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Design problem of π-type noise filter  A circuit diagram of the π-type noise filter to be de-  signed is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The circuit consists of three  elements, namely an inductor and two capacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Fig-  ure 2 shows the π-type noise filter model utilized in this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' It is assumed that the back side of the substrate is  grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The performance of a noise filter is determined  by the capacitance of the capacitor, the inductance of the  inductor, inductive noise, and parasitic capacitance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The  inductive noise and parasitic capacitance depend on the  relative location of the inductor, the capacitors, and the  conductor path, which does not appear in the circuit di-  agram but should be designed as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Substrate  Inductor  Capacitor 1  Capacitor 2  Conductor  Output port  Input port  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Example of element and conductor arrangement for  π-type noise filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The input and output ports, capacitors,  and inductor are represented by simple square elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The  backplane is the electrical ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  x  y  Data acquisition and learning  Hidden true system  y = f(x)  y = xTAx  (1)  (2)  (3)  ~  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Schematic diagram of BOCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' (1) Data y for input  x is obtained from simulation or experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  (2) Second-  order regression equation is estimated from input x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  ˜y is estimated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  (3) Optimal x is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Here, A is  the coefficient of the quadratic regression equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' f is an  unknown function under the governing equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Black-box optimization of noise filter  The objective of black-box optimization is to obtain  the input parameter x that minimizes (or maximizes)  the characteristic value y with a small number of tri-  als under the condition that the relation between x and  y (y = f(x)) is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Here, we focus on Bayesian  Optimization of Combinatorial Structures (BOCS) [16],  which is a learning method applicable to cases where the  input parameter x is a binary variable, as done in the  literature [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In BOCS, the relation between x and y  is learned sequentially using a quadratic regression equa-  tion of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  In other words, starting with several data  sets of x and y, we (1) obtain the data y for the input  x through simulations or experiments on a real system  where the input-output relation is unknown, (2) learn  the relation between data y and input x in quadratic  form, and (3) search for the optimal input x under the  assumed quadratic relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The relation between the  various tasks in BOCS is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  To apply this black-box optimization to the design of  noise filters, we define a binary variable x that specifies  3  Input port  positions  Output port  positions  Capacitor 1  positions  Capacitor 2  positions  Inductor  positions  A  B  C  X  Y  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Candidate element positions and conductor paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  As an example of conductor paths, three candidates (A, B,  and C) between the upper side of the input port and the left  side of capacitor 1 are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  the location of the element and the conductor path, and  employ electromagnetic field analysis using the finite el-  ement method as the data acquisition method in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In  (3), quantum annealing is employed to find the global  minimum in the regression model, which has many lo-  cal minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The solution x of the quantum annealing  and the corresponding output value y are added to the  data in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' We refer to this method  as BOCS-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' To clarify the effect of quantum anneal-  ing, a calculation using simulated annealing (BOCS-SA)  instead of quantum annealing is also performed and the  results are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The following sections describe the binary design vari-  ables that represent electrical component positions and  conductor paths and the characteristic values for evalu-  ating filter performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Binary design variables  The element positions and conductor paths between  the elements are mapped to the binary variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In  this study, the problem is to select the positions of five  elements (an input port, an output port, an inductor, and  two capacitors) from two candidates and the conductor  paths from three candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In order to represent these  variables as binary variables, the substrate is divided into  a 10×15 (X×Y) grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The input and output ports are  placed on the sides of the board and the inductor and  capacitor are placed in the grid as concentrated elements,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Three candidate paths as conductors are created by  connecting the elements from top to bottom in the fol-  lowing manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Draw a path in the X direction and then in the Y  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Draw a path in the Y direction to half of the dif-  ference, then in X, and then in the remaining Y  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Input port �  � Output port  Capacitor 1  Capacitor 2  Inductor  Conductor  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Circuit  corresponding  to  bit  string  “0101101010010001100100” in one-hot representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Draw a path in the Y direction and then in the X  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The typical π-type noise filter, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 2, is appro-  priately included as a candidate by the above conductor  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The present method can be simply extended to  the case with more than three candidate paths if neces-  sary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  We adopt one-hot encoding to represent noise filters  in which element positions and conductor paths are se-  lected from these candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In the case considered here,  22 bits are required because there are two candidates for  each of the five element positions and three candidates  for each of the four conductor paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Let “10” be the  state in which the element is at the bottom or on the  left and “01” be the state in which it is at the top or  on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Then, let “100” be a conductor path that  first moves in the X direction, “010” be one that turns  in the middle, and “001” be one that first moves in the  Y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The bits that represent the conductor path  follow the element position bits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' that is, the first 10 bits  represent the five element positions and the latter 12 bits  represent the selection of the four conductor paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The  bits that represent the element positions are arranged on  the board in the following order from left to right: in-  put port, capacitor 1, inductor, capacitor 2, and output  port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The conductor paths are similarly arranged in the  following order from left to right: input port - capaci-  tor 1, capacitor 1 - inductor, inductor - capacitor 2, and  capacitor 2 - output port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  For example, a circuit en-  coded by “0101101010010001100100” as binary variable  x is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Obtaining characteristic value  The S-parameter S21 is adopted as the characteristic  value y of the noise filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' S21 indicates the ratio of out-  put power to input power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' When the input power of noise  is p1 and the output power is p2, S21 is expressed by the  following equation,  S21 =  �  |p2|  |p1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  (1)  4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Circuit  corresponding  to  bit  string  “1001011001000001000100”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The  conductor  paths  be-  tween the input power port and capacitor 1 and those  between the inductor and capacitor 2 are not selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' To  avoid disconnection, conductors spread over the board are  assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  We design a noise filter that minimizes S21 under the  given noise voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' p1 and p2 are calculated using finite  element analysis for simulating the electromagnetic field  of the electric circuit model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 2, that is, the  model in which the back of the board is the ground and  the electrical components are lumped-parameter ones on  the surface of the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' A sufficiently large air region is  provided around the board in order to precisely calculate  the induced noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' A scattering boundary condition is  set at the outermost boundary of the air region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Special procedures are required in the following two  cases where the S-parameters are not correctly evaluated  by the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  (I) Element position does not satisfy the one-hot con-  straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  (II) A bit in the conductor path is “000” (the circuit  has a disconnection on the board).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  In case (I), the binary variables are unencodable to a  configuration of a noise filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Given such binary vari-  ables, instead of performing the finite element method,  we calculate y as a penalty according to the following  formula,  y = ybase + λ  5  �  m=1  (x2m−1 + x2m − 1)2 ,  (2)  where ybase is the base value of the violation of one-  hot constraints, λ is the penalty coefficient, and xi is  the value of the i-th bit of the binary variable x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Since  BOCS learns characteristic values in quadratic form, this  penalty of one-hot constraints is also expected to be  learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  In case (II), a meaningful S-parameters for evaluating  a noise filter performance cannot be obtained because  the conductor path is disconnected such that voltage is  conducted neither from a normal signal nor noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' We  assign a dummy conductor that avoids the disconnection,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Note that when multiple conductor  paths are selected, such as “011”, we take the sum of the  conductor paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  To summarize, we calculate the characteristic value of  a noise filter y using the following equation,  z ≡  5  �  m=1  (x2m−1 + x2m − 1)2 ,  (3)  y =  �  S21  ( for z = 0 ),  ybase + λz  ( for z ̸= 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  (4)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Parameters for circuit model and black-box  optimization  For the calculation of characteristic values, the sub-  strate thickness, width, and height are set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='6, 150,  and 100 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' An air area of 30 mm is pro-  vided around the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Scattering boundary conditions  are set at the outermost boundaries of this air region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The substrate is divided into a 10×15 grid, as introduced  in section II B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The physical constants of the power supply port, ca-  pacitor, and inductor are set to 50 Ω, 100 F, and 10 H,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The substrate’s relative permittivity, rela-  tive permeability, and conductivity are set to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='5, 1, and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='0 × 10−8 S/m, respectively, assuming an FR-4 sub-  strate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The conductor is treated as a perfect conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  In addition, S21 was calculated using a frequency anal-  ysis at 10 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' For Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' (2), we set ybase = −60 and  λ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The quantum annealer was Advantage system4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='1 by  D-Wave Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  We adopted the Python library  dwave-neal by D-Wave Systems as a simulated anneal-  ing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The sampling number was set to 3000 when  solving the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The x value that gave the small-  est y was adopted as the next candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' For the initial  training datasets of BOCS, we prepared 20 randomly gen-  erated binary variables x and their corresponding char-  acteristic values y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' BOCS-QA and -SA were performed  until 300 searches were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  RESULTS AND DISCUSSION  We compare the results of the BOCS-QA and BOCS-  SA calculations with those of random search in which  binary variables were randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  First, the results of all search histories of BOCS-QA  and random search are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 7 and 8, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The learning processes of BOCS-QA and random  search are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Figure 7 shows that BOCS-QA  mainly learned the penalty term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' (2) in the be-  ginning (before approximately 60th search), and sub-  sequently started to learn on the bases of the perfor-  mance of the noise filter S21, suggesting that the design  of the penalty term facilitated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Then, the high-  est record of S21 was steadily set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' On the other hand,  5  120  100  80  60  40  20  0  0  50  100  150  200  250  300  y  Number of searches  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Full search history of BOCS-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  120  100  80  60  40  20  0  0  50  100  150  200  250  300  y  Number of searches  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Full search history of random search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  the random search shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 8 searched for a feasi-  ble noise filter in very rare cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' There is no particular  trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The number of solutions that satisfy the one-hot  constraint is ten, which is close to the expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The probability that a random binary variable satisfies  the one-hot constraint is 25/210 = 1/32, so the expected  number for 300 searches is nine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The update records of the characteristic value y ver-  sus the number of searches are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Since  BOCS-QA, BOCS-SA, and random search are random-  ized algorithms, the mean, minimum, and maximum val-  ues were calculated for ten trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' BOCS-QA and BOCS-  SA steadily search for a noise filter with good perfor-  mance, whereas random search tends to have a large  variance (especially with a small number of searches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The steady performance improvement of BOCS-QA and  BOCS-SA shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 9 is due to the successful learn-  ing of S21, as confirmed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  At 300 searches,  BOCS-QA shows slightly better performance than that  of BOCS-SA in terms of the average, minimum, and max-  imum values, as shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Next, we evaluate the filter performance of the ob-  tained solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Since there are 222 cases (expressed in  22 bits), enumerating the performance of all solutions is  unrealistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' We therefore choose only the relevant solu-  tions with unique element positions and a single conduc-  tor path between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' This gives a total of 2592  cases (25 = 32 combinations of element positions and  34 = 81 combinations of conductor positions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Figure  Best record of y  Number of searches  search  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Updated records of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The solid and dotted lines  represent the mean and the filled area represents the area  between the maximum and minimum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  0  50  100  150  200  250  300  350  400  450  [–82, –84)  [–84, –86)  [–86, –88)  [–88, –90)  [–90, –92)  [–92, –94)  [–94, –96)  [–96, –98)  [–98, –100)  [–100, –102)  [–102, –104)  [–104, –106)  [–106, –108)  [–108, –110)  [–110, –112)  Frequency  S 21 (dB)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Histogram of S21 value in decibels when element  positions are specified uniquely and there is one conductor  between elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  10 shows a histogram of the S21 value in decibels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' For  our settings, noise filters whose S21 is under −108dB are  rare (approximately 3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Since the average records of  BOCS-QA and BOCS-SA are in the top 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='8% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='9%,  respectively, as shown in Table I, these methods finding  such filters in 300 searches are considered efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The configuration of the best-performing noise filter  obtained using BOCS-QA is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In this  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Comparison of results obtained by various meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Method  Object  Value  Rank  QA  Best  −111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='34 dB  1st  Average −109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='64 dB  19th  Worst  −106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='97 dB 192nd  SA  Best  −110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='55 dB  14th  Average −108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='91 dB  48th  Worst  −104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='80 dB 528th  Random  Best  −107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='12 dB 180th  Average −104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='80 dB 192th  Worst  −102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='00 dB 1058th  6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Noise filter obtained by BOCS-QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  case, the value of S21 was −111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='34 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The input port  and capacitor are placed close to each other, prevent-  ing performance degradation due to induced noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' This  shows that the obtained configuration is physically rea-  sonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  In this study, we formulated a problem with two candi-  dates for the element positions and three candidates for  the conductor paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' If we considered a large-scale prob-  lem with a larger number of candidates, the probability  of finding a well-posed noise filter by chance using ran-  dom search would be much smaller and the superiority  of BOCS-QA and BOCS-SA would be more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  CONCLUSION AND OUTLOOK  To find input parameters that provide the desired char-  acteristics with a small number of searches, we proposed  an iterative optimization method that incorporates quan-  tum annealing in the BOCS framework and applied it to  the problem of designing noise filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' A π-type noise fil-  ter that consists of two capacitors and an inductor was  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' A model was created to select two candidates  for the location of these elements and three candidates  for the path of the conductor connecting the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The results show that a high-performance noise filter  can be efficiently found and that the search progresses  more stably than does random search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' This shows that  the framework that incorporates quantum annealing into  black-box optimization is applicable to electric circuit de-  sign problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' The present method could help engineers  meet the high demand for electrical products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  Beyond  the  optimization  of  electric  components  demonstrated here, system-level optimization of electric  devices is a topic for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' It could lead to mul-  tiphysics optimal design that requires simultaneous opti-  mizations of multiple phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  The proposed BOCS framework was proven to work  with quantum annealing and simulated annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content='  A  comparison of these two versions showed only a slight  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' A recent study that compared the two solvers  in an black-box optimization framework also concluded  that clear performance improvements using quantum an-  nealing are rare [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' However, a clear advantage of quan-  tum annealing in finding optimal solutions, achieved by  adjusting the annealing schedule, has recently been re-  ported [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' Future research should thus examine in de-  tail the scheduling protocols to further improve the per-  formance of BOCS with quantum annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
+page_content=' In addition,  a recent improvement of the learning process [17] could  be integrated into the present BOCS framework to speed  up the whole optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE2T4oBgHgl3EQfNQYC/content/2301.03733v1.pdf'}
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1dAyT4oBgHgl3EQfbvcu/content/tmp_files/2301.00267v1.pdf.txt

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+What Promotes Smectic Order: Applying Mean Field Theory to the Ends
+David A. King∗ and Randall D. Kamien
+Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd St., Philadelphia, PA, 19104.
+(Dated: January 3, 2023)
+Not every particle that forms a nematic liquid crystal makes a smectic. The particle tip is critical
+for this behaviour. Ellipsoids do not make a smectic, but sphero-cylinders do. Similarly, only those
+N-CB alkylcyanobiphenyls with sufficiently long (N ≥ 8 carbons) alkane tails form smectics. We
+understand the role of the particle tip in the smectic transition by means of a simple two-dimensional
+model. We model sphero-cylinders by “boubas” with rounded tips, and ellipsoids by “kikis” with
+pointed tips. The N-CB molecules are modelled by a small body with a polymer tail. We find
+that rounded tips and longer polymer tails lead to a smectic at lower densities by making the space
+between layers less accessible, destabilizing the nematic.
+I.
+INTRODUCTION AND FORMULATION
+Onsager recognized that the geometry of particles af-
+fects the structure of their ordered phases [1].
+The
+most remarkable thing about his insight is that the ne-
+matic phase is unremarkable: any fluid of sufficiently
+anisotropic particles will form a nematic liquid crystal,
+where the particles are homogeneously distributed but
+have a preferential orientation.
+However, not all such
+particles form a smectic-A phase, a phase with the same
+orientational order but with a periodic density modula-
+tion in the direction of alignment. This was noticed by
+Frenkel [2, 3], who considered a system of parallel ellip-
+soids. Smectics have strong orientational order, so the
+particles may be assumed parallel without loss of gener-
+ality. He argued that this system had no smectic phase
+because it could be mapped to a system of hard spheres
+in a way that preserves the thermodynamic properties
+by simply rescaling the lengths and momenta parallel to
+the ellipsoids. Hard spheres are only observed to exist in
+fluid or crystalline phases, so the ellipsoids can have no
+smectic phase.
+This argument is extremely elegant, but leaves some
+open questions; what if the particle shape is only ap-
+proximately an ellipsoid so that the rescaling does not
+produce spheres? Are ellipsoids the only elongated par-
+ticles that miss the smectic phase due to this symmetry?
+Sphero-cylinders have been observed in simulations to
+make smectics [4]; what do they have that those parti-
+cles without smectic phases do not? Some of these ques-
+tions can be tackled using density functional theory and
+similar methods [5–11], but this often results in compli-
+cated analyses and it is difficult to gain insight into the
+differences between different particle shapes.
+It is useful to look for another instance of two molecules
+with similar structures where one has a smectic phase but
+the other does not, so that similarities to the case of ellip-
+soids and sphero-cylinders can be sought. Such an exam-
+ple exists and is well known to experimentalists; N-CB
+type alkylcyanobiphenyls [12, 13]. The precise structure
+∗ [email protected]
+of these molecules is shown in Fig.(2), but it is most use-
+ful to think about them as a small “body” to which a
+“tail” made of N links is attached. When N = 8, the
+molecule is a typical thermotropic liquid crystal former,
+and has both a nematic and a smectic-A phase. With
+N = 5, however, the smectic is absent (indeed, for N < 8
+there is no smectic, though most experiments focus on
+5-CB). The common difference between the particles in
+the N-CB example and Frenkel’s case is the structure at
+their ends; their “tips”. This points to the key question
+we would like to answer: why are the particle tips impor-
+tant for the formation of a smectic phase? We will argue
+here that the nematic phase is suppressed by rounded
+tips allowing the smectic phase to intervene.
+This ef-
+fect is similar to the situation found in [14] where the
+introduction of small platelets suppressed the uniaxial
+nematic phase allowing for the onset of the biaxial ne-
+matic. In short, when the mesogen tips are pointed, a
+test mesogen can more easily be inserted between exist-
+ing smectic layers compared to a round-tipped mesogen.
+As a result pointy mesogens more easily fill in the space
+between smectic layers resulting in the nematic phase.
+We tackle this problem by means of a toy model,
+which captures the essential physics but is simple enough
+to be understood fully.
+For this model to be satisfac-
+tory and consistent, it should be able to describe the
+isotropic-nematic (I-N) transition and nematic-smectic
+(N-S) equally well. An (almost) exactly solvable model
+for the I-N transition was developed by Onsager [1], and
+we might start there for inspiration. Onsager’s approach
+relied on the virial expansion which, fortuitously, could
+be truncated. This is because, for highly anisotropic par-
+ticles, the I-N transition happens at rather low concen-
+trations. For the N-S transition this is not the case, and
+the virial expansion breaks down [15]. Hence, we must
+take a significantly different starting point for our model
+that can incorporate interactions between large numbers
+of molecules without appealing to the virial expansion.
+Recall the bouba-kiki effect where, across cultures and
+languages, the word “bouba” is associated with rounded
+shapes and “kiki” with pointed shapes [16–18]1. We will
+1 This effect was first realized by K¨ohler for shapes named
+arXiv:2301.00267v1  [cond-mat.soft]  31 Dec 2022
+
+2
+FIG. 1.
+Sketches of the particle shapes we consider. On the
+left is a “bouba”, with a rectangular mid-section of width w0
+and semi-circular tips of radius w0/2. The “kiki” is on the
+right, whose midsection is the same as the bouba, but whose
+tip is a triangle of height w0/2. Both particles are of total
+length ℓ.
+FIG. 2.
+The chemical structure of the N-CB molecules (
+specifically 8-CB) is given above our crude model for it. We
+think of these molecules as having a small body of size w0 and
+a polymer tail of length lp.
+argue that nature has a similar bias and expresses it by
+allowing boubas to form smectics more easily than kikis.
+To keep the model as simple as possible, we restrict our
+attention to two dimensions and simplify the particle
+structures. We consider “kikis” instead of ellipsoids, and
+“boubas” instead of sphero-cylinders. Both the bouba
+and the kiki are of total length l, and have rectangular
+mid-sections with widths w0, but their tips are differ-
+ent: The boubas have semi-circular tips, of radius w0/2,
+whereas the kikis have triangular tips whose height is also
+w0/2. These are sketched in Fig.(1). We model the N-CB
+molecules, with the same spirit of simplicity, as particles
+with a small body from which a flexible polymer tail of
+length lp emerges. For the cases of interest, 5-CB and
+8-CB, the tail is relatively short, since it only includes
+a few repeating units. This makes the flexible polymer
+a crude model for the tail, as it assumes a very large
+“maluma” (rounded) and “takete” (pointed), although it is most
+famous now with the names bouba and kiki.
+number of monomers. Another simplifying but crude ap-
+proximation we make is to ignore the size of the body, so
+that it has no excluded volume. Nevertheless this should
+not change the physics at the particle tips, which is our
+focus.
+Our approach is built on a simple construction of the
+free energy, which considers one test particle in a given
+background. By supposing that the dominant interaction
+between the particles is their excluded volume, we may
+understand the background as restricting the position of
+the test particle to a particular region. The size of this
+region controls the free energy. This allows interactions
+between large numbers of particles to be accounted for
+qualitatively in much the same way as successful tube
+theories in polymer physics [19] or free volume theory
+[20]. We briefly outline this construction before show-
+ing how it is consistent with virial theory for a simple
+model of the I-N transition. We then apply it to the N-
+S transition for boubas and kikis and, subsequently, N-
+CB molecules. Our calculations demonstrate that boubas
+form smectics at lower densities than kikis, because the
+tip geometry destabilizes the nematic phase. The same
+conclusion applies to the N-CB particles with long tails;
+(N+1)-CB makes a smectic at a lower density than N-
+CB.
+In this missive, we employ a general construction for
+the free energy that has been used before to determine
+the free energy of polymers subject to topological con-
+straints [21, 22]: posit a test particle in state T placed in
+a background in the state B. Later we will give specific
+examples of these states, for example one can imagine T
+to indicate if the test particle is in a “nematic state” or
+a “smectic state”, for example. Assuming that the test
+particle is confined to a given region by the background
+allows us to determine the probability of realizing the
+test particle in some state, given the state of the back-
+ground. We write this conditional probability as P(T |B).
+The probability of realizing the background state, P(B)
+determines in what phase the system lies. For the pur-
+poses of our construction, we suppose it is known and is
+determined by minimizing the free energy.
+We calculate the free energy of the system from Gibbs’
+definition
+βF =
+�
+B
+�
+T
+P(T ∩ B) log P(T ∩ B)
+(1)
+where P(T ∩ B) is the probability of realizing T and
+B, and the sums run over all possible states. Applying
+the identity P(T ∩ B) = P(B)P(T |B) and noting that
+�
+T P(T |B) = 1 we find
+βF =
+�
+B
+P(B) log P(B) +
+�
+B,T
+P(B)P(T |B) log P(T |B)
+(2)
+The first term is understood as the free energy of the
+background, βFB, and the second as the free energy of
+a test particle in a given background, βFT (B), averaged
+over all realizations of that background. The total free
+
+wo
+wo3
+energy of the system is
+βF = βFB + ⟨βFT ⟩
+(3)
+with angle brackets denoting an average over the back-
+ground.
+II.
+ISOTROPIC-NEMATIC TRANSITION
+Let us demonstrate how this construction can be used
+to study liquid crystal transitions by applying it to the
+simplest model of the I-N transition [15]. This involves
+a two-dimensional gas of rods (rectangles) which can
+only be oriented vertically or horizontally. The rods in-
+teract exclusively via their excluded volume, and it is
+supposed that each accesses every allowed position with
+equal probability. In this model, the isotropic phase is
+when the rods are vertical or horizontal with equal proba-
+bility and the nematic when there is a bias one way or the
+other. Zwanzig studied, via a virial expansion, a three-
+dimensional version of this model where the rods can only
+point along the co¨ordinate axes [23]; it can be specialized
+to two-dimensions where the analysis is relatively simple
+[15]. Here we demonstrate that our approach yields the
+same results as the more traditional approach but it also
+allows us to consider densities beyond which the virial
+expansion fails.
+The first step is to define the test particle and back-
+ground states.
+The state of the test particle is deter-
+mined by both its position and orientation, so we write
+T = (T, r).
+Here r is its position and the variable T
+indicates if it is vertical (V ) or horizontal (H). For the
+background, we suppose that every particle is in the same
+orientation, given by the variable B. To completely spec-
+ify the state, we then need to keep track of the positions
+of all the particles {ri} and we write B = (B, {ri}).
+Next we need the conditional probability P(T |B).
+Given our assumptions, we have
+P(T |B) = αT ΘT B(r, {ri})
+(4)
+Here ΘT B(r, {ri}) is a unit indicator function which picks
+out the allowed positions r of a test particle with orien-
+tation T in a background of particles with orientation B
+and positions {ri}. The constant αT , which depends on
+the test particle orientation, is determined by ensuring
+P(T |B) is appropriately normalised. If the probability
+of the test particle being vertical is p, then
+P(V, r|B) =
+p
+ΩV B
+ΘV B(r, {ri})
+(5a)
+and,
+P(H, r|B) = 1 − p
+ΩHB
+ΘHB(r, {ri})
+(5b)
+for the two possible orientations and
+ΩT B({ri}) =
+�
+dr ΘT B(r, {ri})
+(6)
+are normalization factors. Using these expressions we can
+directly compute βFT (B) from (2)
+βFT (B) = βF0(p) − p log ΩV B({ri})
+− (1 − p) log ΩHB({ri})
+(7)
+where βF0(p) = p log p+(1−p) log(1−p) is the standard
+entropy of mixing.
+What do we choose for P(B)? The state B = (B, {ri})
+is realized with probability P(B) = ϕ(B)ψ({ri}), with ϕ
+being the orientational probability and ψ the probability
+of the background particle positions. Both are taken to
+be independently normalized. Next we make the “mean-
+field-like” approximation to say that the probability of
+the background being vertical is the same as that prob-
+ability for the test particle, i.e. ϕ(V ) = p. The same
+is of course true for the probability of being horizontal.
+Putting this into (2) the total free energy as a function
+of p is
+βF(p) = 2βF0(p) − p2 ⟨log ΩV V ⟩ − (1 − p)2 ⟨log ΩHH⟩
+− p(1 − p) (⟨log ΩV H⟩ + ⟨log ΩV H⟩)
+(8)
+where ΩV V is the accessible area to a vertical test par-
+ticle in a vertical background, ΩV H is that for vertical
+test particle in a horizontal background, and so forth.
+The angle brackets denote averaging over all positions
+of the background particles. This expression is simpli-
+fied greatly by noting symmetries of the accessible areas,
+namely
+ΩV V = ΩHH ≡ Ω∥
+and
+ΩV H = ΩHV ≡ Ω⊥
+(9)
+It follows that the free energy is, up to a constant,
+βF(p) = 2βF0(p)−2p(p−1)
+��
+log Ω∥
+�
+− ⟨log Ω⊥⟩
+�
+(10)
+Note the factor of two appearing in front of the entropy
+of mixing term, βF0. This arises because, by artificially
+splitting the system into the test particle and the back-
+ground, we are essentially considering two separate pop-
+ulations of particles. As we shall see shortly, this factor
+of two is correct and leads to the same result as the virial
+approach.
+To explore the I-N transition, we must find the equi-
+librium probability of the system being vertical, p∗, by
+minimizing F(p):
+βF ′(p∗) = 0 = 2 log
+p∗
+1 − p∗ − 2(2p∗ − 1)∆S
+(11)
+where ∆S = ⟨log Ω∥⟩−⟨log Ω⊥⟩. Evidently, when the two
+accessible areas, Ω∥ and Ω⊥, are both equal the only solu-
+tion is p∗ = 1/2. This is always a solution but, depending
+on ∆S, this is not the minimum of the free energy. The
+difficult part of this approach is computing ∆S as a func-
+tion of the density of the system. We will discuss this in
+more detail for the N-S transition but for now, guided
+
+4
+by the knowledge that the I-N transition occurs at low
+density, we make a simple approximation valid in that
+limit. Namely, we employ free volume theory. The test
+particle may access the whole area of the system, A, ex-
+cept those parts where it overlaps with any background
+particle. For sufficiently low densities, the background
+particles all independently exclude some area that does
+not depend on their position.
+Denoting this excluded
+area as aexc
+∥,⊥ in either the parallel or perpendicular case
+we may write, Ω∥,⊥ = A − Naexc
+∥,⊥, and it follows that for
+small area density ρ = N/A:
+∆S = log
+�1 − ρaexc
+∥
+1 − ρaexc
+⊥
+�
+≈ ρ
+�
+aexc
+⊥ − aexc
+∥
+�
+(12)
+Using this in (10) yields the same equation for p∗ as would
+be derived using Onsager’s virial expansion approach
+[15]. This demonstrates the consistency of our construc-
+tion with more traditional approaches for studying liquid
+crystal transitions. The advantage of our method is that
+the free energy is written in terms of the area accessi-
+ble to a single particle. This is relatively straightforward
+to calculate (or estimate) even for concentrated systems
+where the virial expansion breaks down. As we shall see,
+this allows us to study the N-S transition in much the
+same way as the I-N transition.
+III.
+NEMATIC-SMECTIC TRANSITION
+An appealing aspect of our treatment of the I-N tran-
+sition was that the continuous range of orientations a
+real particle can access was replaced by two discrete op-
+tions; vertical and horizontal. To get this simplicity to
+carry over to the study of the smectic phase, we want to
+split the continuous range of positions into two distinct
+choices.
+The defining feature of the smectic phase is that the
+particles lie in distinct layers with a given separation. Let
+us say that these layers are all parallel to the x-axis and
+are separated by h. If our particles have total length ℓ,
+then we must have ℓ < h < 2ℓ, for the layers to make
+sense.
+By analogy to the vertical-horizontal two state
+model of the I-N transition, let us suppose that there
+are two sets of such layers, “solid” and “dashed”. The
+spacing between layers of the same type is h, but the
+layers are interleaved so that the distance between a solid
+and a dashed layer is h/2. The particles can be placed on
+either a solid or a dashed layer. Our goal is to find the free
+energy as a function of p, the probability that a particle
+occupies a solid layer, and to determine the equilibrium
+value p∗. When p∗ ̸= 1/2 we have a smectic-A phase,
+and we identify the state when p∗ = 1/2 as the Nematic.
+Why should this be the case when there is still vertical
+layering?
+To see this, let us consider the definition of
+the smectic order parameter, S [2]. The density of the
+particles as a function of y can be expanded as a Fourier
+FIG. 3.
+Sketches of the smectic and nematic phases in our
+model. In both panels the solid and dashed sets of lines are
+shown. In panel (a) the solid lines are preferred to the dashed
+by the particles, i.e p ̸= 1/2. This is the smectic phase. Panel
+(b) has the solid and dashed lines occupied equally, p = 1/2.
+This is the nematic.
+series
+ρ(y) − ¯ρ =
+∞
+�
+n=1
+ρn cos (2πny/h + δn)
+(13)
+where the n = 0 mode defines the average density, ¯ρ,
+there is an arbitrary phase per mode, δn, and h is the
+aforementioned layer spacing. The coefficient of the n =
+1 mode defines the smectic order parameter, S ≡ ρ1. The
+nematic and smectic phases in this model are sketched in
+Fig.(3). When the solid and dashed layers are occupied
+with equal probability it is clear that
+ρ(y) − ¯ρ = ρ2 cos
+�
+4π y
+h + δ2
+�
++ · · ·
+(14)
+hence S = 0 identically in this case. While there is now a
+new smectic with half the periodicity of the target phase,
+that is not the smectic for which we are looking! This is
+why we identify this as the nematic phase, even though
+there is a “higher level” layered order present. This sit-
+uation is likewise true for the two-state model of the I-N
+transition: when vertical and horizontal orientations are
+equally likely, the nematic order parameter vanishes, but
+there is still 4-fold orientational order in the system.
+We construct the free energy as a function of p using
+the same test particle and background construction as
+before. The state of the test particle, T = (T, x), tells us
+both whether it sits on a solid or dashed line and its x-
+position on that line and T = S when it is on a solid line
+and T = D when on a dashed line. We assume that all
+allowed x-positions of the test particle, not overlapping
+with a background particle, are equally likely.
+For the background state, B, all of the particles occupy
+the same set of layers; either they are all on solid or all
+on dashed. We also need to keep track of the x-positions
+of all of the particles. This may appear intimidating, but
+notice that we need only keep track of those particles on
+layers which interact with the test particle, because all
+of the others will drop out of the calculation. We refer to
+the set of x-co¨ordinates for these particles by {xi}, the
+range of the index i depends on with how many layers
+the test particle interacts. Again B = S for solid and
+
+5
+FIG. 4.
+An example state of the test particle (picked out in
+red) and the background. Here, the test particle is in state
+T = (D, x), sitting on a dashed layer. The background is in
+state B = (S, {xi}), with all particles on solid layers. The
+set of co¨ordinates {xi} are the x-positions of the background
+particles. Only a selection of the background particles closest
+to the test particle are shown.
+B = D for dashed. Furthermore, we may assume that
+each layer of the background has length L and is occupied
+by N particles. We shall call the line density on each
+layer ν = N/L. All together, we write B = (B, {xi}).
+In Fig.(4) we sketch an example state of the background
+and test particle.
+The conditional probability is
+P(T |B) = p(T) ΘT B (x, {xi})
+ΩT B ({xi})
+(15)
+Here, p(T) is the probability of the test particle being
+on a solid T = S line or dashed T = D lined and
+ΘT B(x, {xi}) is a unit selector function picking out when
+the test particle at position x does not overlap with any
+of the background particles. This latter function deter-
+mines the “accessible length” for the test particle and
+provides the proper normalization
+ΩT B ({xi}) =
+� ∞
+−∞
+dx ΘT B (x, {xi})
+(16)
+Define p(T = S) = p for the probability of the test par-
+ticle being on a solid line so p(T = D) = 1− p. Applying
+the same mean-field approximation as we did for the I-N
+transition we choose p(B = S) = p and p(B = D) = 1−p
+and follow the steps that led to (8) to obtain
+βF(p) =2βF0(p) − p2 ⟨log ΩSS⟩ − (1 − p)2 ⟨log ΩDD⟩
+− p(1 − p) [⟨log ΩSD⟩ + ⟨log ΩSD⟩]
+(17)
+Similar symmetries to (9) apply due to the equivalence of
+shifting the whole system along y by h/2 (solid/dashed
+duality);
+ΩSS = ΩDD ≡ Ωo,
+and
+ΩSD = ΩDS ≡ Ωx.
+(18)
+Up to a constant, the free energy is
+βF(p) = 2βF0(p) − 2p(p − 1) [⟨log Ωo⟩ − ⟨log Ωx⟩] . (19)
+Note how similar this is in structure to (10) for the I-N
+transition. Hence, the equation determining p∗ is pre-
+cisely the same as (11):
+log
+p∗
+1 − p∗ = (2p∗ − 1)∆S
+(20)
+where we have defined ∆S = ⟨log Ωo⟩ − ⟨log Ωx⟩. We see
+that when ∆S > 2, a smectic phase forms with p∗ ̸= 1/2.
+So the problem all comes down to computing ∆S for the
+boubas and kikis and the N-CBs – the key here is that
+we do not need to rely upon the low-density limit. In the
+following we will estimate ∆S directly in the spirit of the
+Tonks gas [24]. Note that ∆S is a function of the layer
+spacing, h, the density on each layer ν and the average
+density ¯ρ = Number/Area = N/(Lh) = ν/h. Our aim is
+to show that boubas undergo a N-S transition at a lower
+density than kikis, and to elucidate the difference that
+the tip shape makes. For the N-CBs, we would like to
+show that the larger N is, the lower the density at which
+the smectic forms. We do not aim to precisely determine
+the phase boundary in any case, that would require a
+more sophisticated method.
+A.
+Boubas versus Kikis
+The
+whole
+calculation
+boils
+down
+to
+computing
+⟨log Ωo⟩ and ⟨log Ωx⟩. In the first case, the test parti-
+cle only interacts with those background particles on its
+own layer, because of the restriction h < 2ℓ. This also
+means that the result will be identical for boubas and
+kikis, because the tip geometry is irrelevant when inter-
+acting with mesogens on the same layer. The starting
+point is an expression for Ωo.
+Let x2 be the distance
+between the centers of the closest background particle
+to the left and right of the test particle. The accessible
+length is then simply
+Ωo = x2 − 2w0,
+(21)
+because each background particle excludes a length w0,
+as shown in Fig.(5). So, we must compute
+⟨log Ωo⟩ =
+�
+dx2 P(x2) log(x2 − 2w0),
+(22)
+where P(x2) is the probability of realizing the distance
+x2. Each layer is a Tonks gas [24], a one dimensional
+gas of finite sized particles interacting only via excluded
+volume.
+The distance x2 is the next-nearest-neighbor
+distance for such a gas, and its distribution, P(x2) was
+calculated by Tonks. This allows us to explicitly calculate
+(22).
+This is done in Appendix A, but here we make
+an approximation which make our analysis very simple,
+but does not change the outcome. The approximation
+replaces
+⟨log Ωo⟩ → log⟨Ωo⟩ = log (2/ν − 2w0) ,
+(23)
+where we have used Tonks’ result ⟨x2⟩ = 2/ν.
+Now we turn our attention to ⟨log Ωx⟩. Once again we
+shall replace this with log⟨Ωx⟩, but the complete calcu-
+lation is in Appendix A. In this case, there are no back-
+ground particles on the same layer as the test particle.
+However, the occupied layer above is only vertically sep-
+arated from it by h/2, so it may interact with that layer
+
+6
+FIG. 5.
+A sketch of the test particle (in red) on a solid layer,
+when the background particles are also all on solid layers.
+The two background particles closest to the test particle are
+indicated. These two are separated by a distance x2. Each
+excludes a length of w0 to the test particle, so that the acces-
+sible length to it in this configuration is Ωo = x2 − 2w0.
+and it likewise interacts with the layer beneath. Let us
+refer to the closest background particles on the left and
+right as xL and xR, respectively. We supply these with
+the superscripts a or b to indicate if they come from the
+layer above or below the test particle so that, xa
+L is the
+position of the closest particle on the layer above the test
+particle to its left and so on. Now, we can write Ωx as
+Ωx = min
+i∈(a,b) xj
+R − max
+i∈(a,b) xi
+L − 2w(h)
+(24)
+so that the absolute left and right limits for the test par-
+ticle are set by the background particles closest to it. The
+function w(h) is the length excluded by the particle, its
+effective width, which must be a function of h because of
+the shape of the tip. Note that the function w(h) is dif-
+ferent for different tip shapes. This expression requires
+us to consider the four possible arrangements of back-
+ground particles. One example is for the closest on the
+left to come from the layer above and that on the right to
+come from the layer below. In this situation if we move
+from all the way to the left to all the way to the right, we
+encounter the background particles from different layers
+in the order; below, above, below, above. This situation
+is sketched in Fig.(6). We shall refer to this configura-
+tion as (baba), and all others accordingly. The accessible
+lengths in each case are simply
+(abab) → Ωx = xa
+R − xb
+L − 2w(h),
+(25a)
+(abba) → Ωx = xb
+R − xb
+L − 2w(h),
+(25b)
+(baab) → Ωx = xa
+R − xa
+L − 2w(h),
+(25c)
+(baba) → Ωx = xb
+R − xa
+L − 2w(h).
+(25d)
+FIG. 6.
+The red test particle sits on a solid layer in a
+background of particles on dashed layers.
+The four closest
+background particles to the test particle are shown; two on the
+layer above and two on the layer below. Using the conventions
+of equation (25), this is the configuration (baba).
+Because
+of the shape of the tips, the background particles exclude a
+length of w(h) < w0. The length accessible to the test particle
+is Ωx, as given in (25d).
+By symmetry, all four of these situations are realized
+with equal probability, so that the average ⟨Ωx⟩ over all
+realizations of the background is
+⟨Ωx⟩ = 1
+4
+�
+2⟨xa
+R − xa
+L⟩ + 2⟨xb
+R − xb
+L⟩ − 8w(h)
+�
+(26)
+The angle brackets here denote averaging over all posi-
+tions xa,b
+R,L. Notice that the combinations xa,b
+R − xa,b
+L
+are
+both the nearest neighbor distance in the Tonks gas, x1.
+The average of this is, ⟨x1⟩ = 1/ν so that
+log⟨Ωx⟩ = log (1/ν − 2w(h))
+(27)
+We now have an expression for ∆S, and the condition for
+a Smectic phase is
+∆S = log 2 + log
+�
+1 − νw0
+1 − 2νw(h)
+�
+> 2
+(28)
+This can be cast as a condition on w(h)
+2w(h) > 2
+e2 w0 + 1
+2ν (e2 − 2)
+(29)
+or, assuming that ν is relatively large, a looser condi-
+tion is 2w(h) ≳ w0. This is the result of the more de-
+tailed analysis in Appendix A and is understood simply
+as comparing the length excluded to the test particle by
+the background particles, 2w(h), to that excluded by the
+background to themselves, w0. Crudely speaking, does
+the background allow enough room for the test particle
+to muscle its way in between the layers? Naturally, this
+will depend on the width of the particle’s shoulders ex-
+pressed through its tip geometry. This is quantified by
+understanding the function w(h).
+
+7
+FIG. 7.
+A sketch of two particles on layers separated by
+h/2 colliding at their tips. The particles shown are boubas,
+but the geometry is equivalent for any shape. The symmetric
+tip shape function, s(x), is indicated in orange. The distance
+between the centers of the two particles is shown in green;
+this is the excluded length, w(h). Equation (32) for w(h) is
+found by considering the y-co¨ordinate of the point P where
+the particles meet.
+Consider a generic particle of width w0 whose tip has
+a symmetric shape described by the function y = s(x).
+This function describes the height of the tip above the
+midsection of the particle at a position x along its width.
+We require −w0/2 ≤ x ≤ w0/2, and symmetry enforces
+s(x) = s(−x). We suppose that the full length of the
+particle is ℓ and that the total length of one tip is t. The
+function w(h) is determined by finding the point P, indi-
+cated in Fig.(7), where two oppositely oriented particle
+tips touch if the centers of the particles are vertically
+separated by a distance h/2. Considering only the lower
+particle we have
+P =
+�
+w(h)/2, ℓ/2 − t + s (w(h)/2)
+�
+(30)
+and considering the upper particle we find,
+P =
+�
+w(h)/2, h/2 − ℓ/2 + t − s (−w(h)/2)
+�
+(31)
+These expressions must both represent the same point,
+hence
+2s
+�w(h)
+2
+�
+= h
+2 − ℓ + 2t.
+(32)
+If we know the function s(x) describing the tip shape,
+then we can find w(h).
+For boubas and kikis, s(x) is
+particularly simple.
+A bouba has a semi-circular tip of radius w0/2 so
+t = w0/2 and sB(x) =
+�� w0
+2
+�2 − x2, which leads to
+wB(h) =
+�
+w2
+0 − (h/2 − ℓ + w0)2. For kikis, whose tips
+are triangular with height t = w0/2 and so sK(x) =
+w0
+2 − |x|, and hence wK(h) = ℓ − h
+2 .
+With the condition (A11) along with the functions
+wB(h) and wK(h) we can find conditions for which values
+of h boubas and kikis form smectics. For boubas
+hB ≤ 2l − (2 −
+√
+3)w0 ≈ 2ℓ − 0.27w0
+(33)
+and for kikis
+hK ≤ 2ℓ − w0
+(34)
+Evidently, boubas will form a smectic for a larger layer
+spacing h than kikis.
+Because we can relate h to the
+number density h = ν/¯ρ, this implies that boubas make
+a smectic at a smaller average density ¯ρ than kikis. It is
+essential to note that the entropy difference arises from
+considering test rods that are not on the background
+smectic layer. In this sense, it is the nematic phase that
+is being changed, not the smectic.
+When the tips are
+pointier there is more opportunity for a rod to find space
+in half layer between the smectic layers.
+It is interesting to consider briefly the limiting case
+when the particle tips become flat. Now the particles are
+rectangles with dimensions w0 × ℓ. The effective width
+for these shapes has a step; w(h) = 0 for h ≥ 2ℓ and
+w(h) = w0 for h < 2ℓ. The calculation given above tells
+us that these rectangles form a smectic when the layer
+spacing becomes h < 2ℓ.
+However, applying Frenkel’s
+rescaling argument [2], we can map the rectangles onto
+a system of w0 × w0 squares. We would then say that
+these squares form a smecticas soon as h < 2w0. Noth-
+ing prevents this from happening in principle but such
+a phase is not observed in simulations [25, 26]. Though
+some calculations do predict a smectic phase, it is ex-
+pected to be unstable to fluctuations for infinite systems
+[27]. In our case, when the layer spacing is just larger
+than the transition value 2w0, the system should be “ne-
+matic” with the dashed and solid layers equally occupied.
+Given that these layers are spaced by a little more than
+w0, the squares will be just touching those on the layer
+above or below. In this way, the order in the y-direction
+is the same as would be observed in a crystal but the
+difference between this state and a crystal is the order
+in the x-direction where we have a Tonks gas. It could
+be argued that the instability shown by our calculation
+when the layer spacing is decreased is actually the in-
+stability to forming the crystal. Given that the particles
+can only occupy layers separated by h/2 and h, this in-
+stability will artificially give rise to a smectic phase for
+squares.
+B.
+N-CB Molecules
+Finally, let us consider the N-CB molecules.
+We
+use the same free energy construction as before for the
+boubas and kikis. This time, we must also keep track
+of the degrees of freedom for the test particle and back-
+ground polymer tails. For simplicity we ignore the size
+of the body of the molecule and the self-excluded volume
+of tail. We are lead to exactly the same form of equation
+for p∗ as (20), and exactly the same condition for the
+smectic phase, namely,
+∆Spoly ≡
+�
+log Ωpoly
+o
+�
+−
+�
+log Ωpoly
+x
+�
+≥ 2.
+(35)
+
+P
+1/2
+s(c)
+l/2 -t
+98
+Here log Ωpoly
+o
+is the entropy of the polymer tail of the
+test particle when it sits on a solid line in a background
+of particles on solid lines, and log Ωpoly
+x
+is the entropy
+when the test particle is on a dashed (solid) line and the
+background particles are on solid (dashed) lines. In this
+expression, the angle brackets denote averaging over all
+positions of the background particle bodies and all con-
+figurations of their polymer tails. Just as for the boubas
+and kikis, we assume that the particle density on each
+layer is ν.
+To make progress, we make the same approximation as
+before
+�
+log Ωpoly�
+≈ log
+�
+Ωpoly�
+. In this way, each term
+can be understood as the entropy of the test polymer tail
+in a fixed average background. Due to the excluded vol-
+ume of the background polymer tails, the presence of the
+background acts to restrict the accessible configurations
+of test polymer. A simple model for this is to say that
+the test polymer is confined to a rectangular box with di-
+mensions Lx × Ly. The lengths Lx,y depend on whether
+we consider
+�
+Ωpoly
+o
+�
+or
+�
+Ωpoly
+x
+�
+.
+In the former case, the width in the x-direction is the
+average next-to-nearest neighbor distance in the Tonks
+gas, Lx
+o = 2/ν. The height in the y-direction in this case
+is the distance between the two closest layers to that
+on which the test particle sits, Ly
+o = 2h. In the latter
+case, the width and heights are halved.
+The width is
+the nearest neighbour distance in the Tonks gas Lx
+x =
+1/ν, and, if the test particle is on a dashed (solid) layer,
+the height is the distance between the two closest solid
+(dashed) layers Ly
+x = h.
+It is now a straightforward polymer physics problem
+[19, 28] to compute the entropies of the polymers in these
+boxes. While we can obtain expressions of
+�
+Ωpoly
+o
+�
+and
+�
+Ωpoly
+x
+�
+for any polymer chain length lp (see Appendix
+B), let us focus for now on two important limits; polymers
+much smaller than the boxes, and those much longer. In
+the first instance we must have lp ≪ h, ν−1 and we find
+⟨Ωo⟩ ∼ 2
+ν + O(lp/h),
+and
+⟨Ωx⟩ ∼ 1
+ν + O(lp/h). (36)
+Here, there is no smectic transition since ∆S ≈ log 2 < 2.
+In the second case, where the polymers are long, we
+must have lp ≫ h, ν−1. This leads to
+⟨Ωo⟩ ∼ 26
+π3ν e−l2
+p(ν2+h−2)/4,
+(37a)
+and,
+⟨Ωx⟩ ∼ 25
+π3ν e−l2
+p(ν2+h−2).
+(37b)
+Therefore the smectic condition is
+∆S = log 2 + 3
+4l2
+p
+� 1
+h2 + ν2
+�
+≥ 2.
+(38)
+In the same way as for the boubas and kikis, this can be
+read as a condition on the layer spacing, h. Namely, for
+FIG. 8.
+∆S for plotted a function of the ratio of the polymer
+tail length to the layer spacing, lp/h. The blue curve is ∆S
+and the orange line is the value it must exceed for a smectic
+to form. This happens for lp/h indicated by the red dashed
+line. This critical ratio is less than, but close to, unity.
+a smectic, we must have
+h2 ≤
+� 4
+3l2p
+(2 − log 2) − ν2
+�−1
+∼ l2
+p
+(39)
+So it follows that particles with longer polymer tails form
+a smectic at larger layer spacings than those with shorter
+tails. This implies that they also form at lower densities.
+The limit of very short polymer tails also showed us that
+there are some tails which are so short that they do not
+form smectics at all. The physical reason for these differ-
+ences is essentially the same as that for the boubas and
+kikis; the longer polymer tails make it harder for particles
+to penetrate between the smectic layers.
+We can also plot the full form of ∆S as a function
+of lp/h at fixed density, assuming that ¯ρ = h−2. This is
+shown in Fig.(8). There we see that the smectic condition
+is met for longer polymers, with values of lp/h ≲ 1.
+At this point one might raise concern about our choice
+of box size. While the widths in the x-direction are clear
+enough, there may be some question about the chosen
+heights. The background may be thought of as layers of
+polymer brushes of some height H < h. It is intuitive to
+expect the these brushes prevent the test polymer from
+reaching all the way to the nearest layer, by virtue of the
+excluded volume interactions. To capture this effect, the
+box height should be reduced by an amount proportional
+to the brush height; h → h − αH, where α < 1. The
+brush height depends on lp and ν and, with reference to
+the simple arguments of Alexander [29] and de Gennes
+[30, 31], as well as the more sophisticated results of Mil-
+ner, Witten, and Cates [32], it must increase when lp or
+ν are increased. This modification only serves to make
+shorter polymer tails worse at making smectics compared
+to longer tails. While more involved treatments of the
+polymer tail entropy are possible and will alter the details
+of our conclusions, we do not expect them to change the
+underlying result that, longer polymer tails de-stabilize
+the nematic phase by making the interstices between lay-
+
+△S
+1.5
+1.0
+0.5
+5 p/h
+0.2
+0.4
+0.6
+0.8
+1.09
+ers less accessible.
+IV.
+CONCLUSIONS
+We have explored which particles can form a smectic-A
+phase by means of a simple two dimensional model. In
+this model, we consider a single test particle in a fixed
+background which restricts the positions of the test par-
+ticle to a well defined region. The size of the region de-
+termines the entropy of the test particle and, by means
+of a mean-field-like approximation, the free energy of the
+system. This construction qualitatively includes the in-
+teractions between a large number of particles allowing
+it to be applied to higher density systems for which ap-
+proaches based on the virial expansion are not valid. In
+particular this allows the nematic-smectic transition to
+be treated on the same footing as the isotropic-nematic.
+We demonstrated that our construction is exactly con-
+sistent with virial approaches to the I-N transition in the
+low density limit.
+We considered the N-S transition for two different rigid
+particle shapes and for N-CB molecules. The rigid par-
+ticles chosen were boubas and kikis, shown in Fig.(1).
+These model three dimensional sphero-cylinders and el-
+lipsoids respectively. It has been noted previously that
+ellipsoids do not form a smectic but sphero-cylinders do.
+Similarly it is known that 8-CB forms a smectic while
+5-CB does not. Our model for these molecules is a small
+body with a polymer tail of a given length. It is expected
+then that longer polymer tails lead to smectics at lower
+densities.
+The analysis of our simple model shows that particles
+with “fatter” tips form smectics at lower densities than
+those with “thinner�� ones. The reason for this is that
+fatter tips allow less space between the smectic layers to
+any rogue interloper trying to make a new home away
+from its own layer, thereby de-stabilizing the nematic at
+a given density. This same reasoning applies to the N-
+CB molecules, where it is the longer polymer tails which
+make the region between the smectic layers less accessi-
+ble.
+Of course the approach that we have taken is only
+approximate and will not give accurate predictions for
+the phase boundary. In the same way, we have not ad-
+dressed the smectic-crystal transition. This would com-
+plete the picture by demonstrating that for kikis, say,
+the N-S transition actually happens at a higher density
+than crystallization, but this is beyond the reach of our
+simple model. Due to the reduction of degrees of free-
+dom in two dimensions, the predicted order of the phase
+transitions discussed may be incorrect. In principle our
+approach may be followed in 3D, but this could result in
+sufficiently complicated analyses that our sacrifices made
+in the name of simplicity may not be worthwhile. Never-
+theless, our simple arguments elucidate the physics gov-
+erning which particles can form smectic phases.
+This work was supported by a Simons Investigator
+grant from the Simons Foundation to R.D.K.
+Appendix A: Boubas and Kikis
+Here we compute ∆S, from equations (19) and (20),
+relevant for the N-S transition of boubas and kikis with-
+out the approximation ⟨log Ω⟩ ≈ log⟨Ω⟩.
+The first step is computing ⟨log Ωo⟩. This is given in
+equation (22) in terms of P(x2), the distribution of next-
+nearest neighbour distance in the Tonks gas. This distri-
+bution may be found exactly [24], and is given by
+P(x2) = ν2 (x2 − 2w0)
+(1 − νw0)2
+exp
+�
+−
+ν
+1 − νw0
+(x2 − 2w0)
+�
+(A1)
+for x2 ≥ 2w0 and zero otherwise. Integrating, we have
+⟨log Ωo⟩ = 1 − γ + log 1 − νw0
+ν
+,
+(A2)
+where γ being the Euler-Mascheroni constant [33].
+Next we require ⟨log Ωx⟩.
+As discussed in the main
+text, we need to consider the four cases (25).
+It is
+convenient for us to write these positions in terms of
+xi
+1 = xi
+L−xi
+R, that is the nearest-neighbur distance in the
+Tonks gas in layer i. It is also useful to introduce the sep-
+aration of the closest particles on the left, ∆L = xa
+L −xb
+L.
+Note that, in order for “left” and “right” to make sense,
+we must have |∆L| ≤ xi
+1. This gives us
+(abab) → Ωx = xa
+1 − |∆L| − 2w(h)
+(A3a)
+(abba) → Ωx = xb
+1 − 2w(h)
+(A3b)
+(baab) → Ωx = xa
+1 − 2w(h)
+(A3c)
+(baba) → Ωx = xb
+1 − |∆L| − 2w(h)
+(A3d)
+All of these are realized with equal probability, so that the averaging ⟨log Ωx⟩ over all realizations of the background
+results in
+⟨log Ωx⟩ = 1
+4
+�
+dxa
+1P(xa
+1)
+�
+dxb
+1P(xb
+1)
+�
+d∆LP(∆L)
+�
+log(xa
+1 − 2w(h)) + log(xb
+1 − 2w(h))
++ log(xa
+1 − |∆L| − 2w(h)) + log(xb
+1 − |∆L| − 2w(h))
+�
+(A4)
+
+10
+Notice that the distributions P(xa
+1) and P(xb
+1) are the same and normalized, so that first two terms in the square
+brackets are the same as are the final pair. This leaves
+⟨log Ωx⟩ = 1
+2
+�
+dx1P(x1)
+�
+d∆LP(∆L)
+�
+log(x1 − 2w(h)) + log(x1 − |∆L| − 2w(h))
+�
+(A5)
+To take the integral over ∆L we need its probability distribution. Because the a layer and b layer are independent of
+each other this must be uniform. The only restriction is on its magnitude |∆L| ≤ x1. Hence,
+⟨log Ωx⟩ = 1
+2
+�
+dx1P(x1)
+� x1
+0
+d∆L
+x1
+�
+log(x1 − 2w(h)) + log(x1 − |∆L| − 2w(h))
+�
+(A6)
+and so
+⟨log Ωx⟩ = −1
+2 +
+�
+dx1P(x1)
+� x1
+0
+log(x1−2w(h)). (A7)
+This is now written in an analogous way with (22), only
+now in terms of the distribution of nearest-neighbor sep-
+arations in a Tonks gas P(x1). This distribution was also
+worked out by Tonks [24]
+P(x1) =
+ν
+1 − νw0
+exp
+�
+−
+ν
+1 − νw0
+(x1 − w0)
+�
+(A8)
+This is straightforward, although this time the result is
+not quite as compact,
+⟨log Ωx⟩ = −1
+2 + log 1 − νw0
+ν
++
+� ∞
+0
+dξ e−ξ log(ξ + α)
+(A9)
+with α = ν(w0 − 2w(h))/(1 − νw0). While the ξ integral
+can be written in terms of incomplete Gamma functions
+[33] it is not particularly illuminating.
+Now we have ∆S, and the condition for a stable smec-
+tic phase is
+∆S = 3
+2 − γ −
+� ∞
+0
+dξ e−ξ log(ξ + α) > 2
+(A10)
+The parameter α is a function of both the tip shape,
+and the density.
+Therefore, this inequality relates the
+density for the N-S transition to the tip shape. When the
+integral in this inequality becomes sufficiently negative,
+the inequality is satisfied. The integral is positive for all
+positive α, but becomes infinitely negative when α < 0.
+Thus, given ν ≥ 0 and w0 ≥ 0, the condition required for
+the smectic phase is,
+2w(h) ≥ w0
+(A11)
+This is qualitatively the same as the relation (29) derived
+using the approximations in the main text.
+Appendix B: N-CB Molecules
+Here we compute ∆S for the N-S transition of N-CB
+molecules.
+The approximation ⟨log Ω⟩ ≈ log⟨Ω⟩ is re-
+quired here to avoid a complicated self-consistent treat-
+ment of the polymer. Within this approximation, each
+term in ∆S can be thought of as the entropy of a poly-
+mer in a 2D box with dimensions Lx × Ly.
+Finding
+this entropy is a standard problem [19] and the start-
+ing point is the polymer Green’s function G(x, x′; y, y′|n)
+which solves
+� ∂
+∂n − b2
+6
+� ∂2
+∂x2 + ∂2
+∂y2
+��
+G(x, x′; y, y′|n)
+= δ(x − x′)δ(y − y′)δ(n),
+(B1)
+and is subject to the boundary conditions at the walls of
+the box
+G(x = 0, Lx, x′; y, y′|n) = G(x, x′; |y| = Ly/2, y′|n) = 0.
+(B2)
+Here the co¨ordinates x′ and y′ represent the horizontal
+and vertical positions of the start of the polymer chain.
+Note that x′ may take any value allowed by the box, but
+we require y′ = 0. The variable n represents the number
+of monomers making up the chain and b measures the
+bond lengths between monomers. The entropy can be
+computed via
+Ω(Lx, Ly) =
+� Lx
+0
+dx
+� Lx
+0
+dx′
+� Ly/2
+−Ly/2
+dy G(x, x′; y, y′ = 0|n).
+(B3)
+The Green’s function is found by separation of variables G = gx(x, x′|n)gy(y|n), with
+gx(x, x′|n) = 2
+Lx
+∞
+�
+m=1
+sin
+�mπx
+Lx
+�
+sin
+�mπx′
+Lx
+�
+exp
+�
+−m2 π2nb2
+6L2x
+�
+,
+(B4a)
+
+11
+and
+gy(y|n) = 2
+Ly
+∞
+�
+m=0
+cos
+�(2m + 1)πy
+Ly
+�
+exp
+�
+−(2m + 1)2 π2nb2
+6L2y
+�
+.
+(B4b)
+Identifying the length of the polymer chain as l2
+p = π2nb2/6 and taking the integrals in (B3) we find
+Ω(Lx, Ly) = 25
+π3 Lx
+�
+p∈Odd
+∞
+�
+m=0
+(−1)m
+p2(2m + 1) exp
+�
+−l2
+p
+� p2
+L2x
++ (2m + 1)2
+L2y
+��
+.
+(B5)
+Taking the limit that the polymer is much smaller than the box, lp ≪ Lx, Ly yields
+Ω(Lx, Ly) ∼ Lx,
+(B6)
+For the opposite limit lp ≫ Lx, Ly we find
+Ω(Lx, Ly) ∼ 25
+π3 Lx exp
+�
+−lp
+� 1
+L2x
++ 1
+L2y
+��
+.
+(B7)
+These expressions reduce to (36) and (37) of the main text when the appropriate box dimensions are used.
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+Bimanual Telemanipulation with Force and Haptic Feedback through an
+Anthropomorphic Avatar System
+Christian Lenz∗, Sven Behnke
+Institute for Computer Science VI, Autonomous Intelligent Systems, University of Bonn, Friedrich-Hirzebruch-Allee 8, 53115 Bonn,
+Germany
+Abstract
+Robotic teleoperation is a key technology for a wide variety of applications. It allows sending robots instead of humans
+in remote, possibly dangerous locations while still using the human brain with its enormous knowledge and creativity,
+especially for solving unexpected problems. A main challenge in teleoperation consists of providing enough feedback to
+the human operator for situation awareness and thus create full immersion, as well as offering the operator suitable control
+interfaces to achieve efficient and robust task fulfillment. We present a bimanual telemanipulation system consisting of
+an anthropomorphic avatar robot and an operator station providing force and haptic feedback to the human operator.
+The avatar arms are controlled in Cartesian space with a direct mapping of the operator movements. The measured
+forces and torques on the avatar side are haptically displayed to the operator. We developed a predictive avatar model
+for limit avoidance which runs on the operator side, ensuring low latency. The system was successfully evaluated during
+the ANA Avatar XPRIZE competition semifinals. In addition, we performed in lab experiments and carried out a small
+user study with mostly untrained operators.
+Keywords:
+force feedback control, teleoperation, dual arm manipulation, human robot interaction, real-time control,
+haptic interface
+1. Introduction
+Published in Robotics and Autonomous Systems, 2022 https://doi.org/10.1016/j.robot.2022.104338
+Teleoperation is a very powerful method to control
+robots.
+It enables humans to explore remote locations
+and to interact there with objects and persons without be-
+ing physically present. Although state-of-the-art methods
+for autonomous control are improving rapidly, the expe-
+rience and instincts of humans, especially for solving un-
+predictable problems is unparalleled so far. The current
+COVID-19 pandemic is a great example of scenarios where
+remote work is highly desirable. Further possible appli-
+cations for teleoperation include disaster response where
+humans can operate remotely and use critical situation-
+saving skills without risking their lives as well as main-
+tenance and healthcare to allow experts operating in re-
+mote locations for manipulation tasks without the need of
+travel. Robotic teleoperation is a popular research area
+which is advanced by multiple robotic competitions like
+the DARPA Robotics Challenge [1] and RoboCup Res-
+cue [2]. These events are a great opportunity to bench-
+mark and evaluate different highly integrated and complex
+systems in standardized test scenarios under comparable
+conditions. Our team NimbRo participates in the ANA
+∗Corresponding author
+Email addresses: [email protected] (Christian Lenz),
+[email protected] (Sven Behnke)
+Avatar XPRIZE Competition1 with the goal to advance
+the state of the art of such robotic telemanipulation and
+telepresence systems.
+In addition to immersive visualization of the remote
+location, one important aspect is telemanipulation which
+enables the operator to physically interact with the re-
+mote environment.
+This capability is critical for many
+applications—without it, we are constrained to mere telep-
+resence.
+In this work, we present a humanoid bimanual tele-
+manipulation system built from off-the-shelf components,
+which allows a human operator to interact and manipu-
+late in remote locations (see Fig. 1). Our contributions
+include:
+1. Integrating a bimanual robotic avatar and an upper-
+body operator exoskeleton for Cartesian telemanip-
+ulation,
+2. an arm and hand controller with force and haptic
+feedback,
+3. a model-based arm movement prediction to hapti-
+cally display position and velocity limitations of the
+remote avatar in real time,
+4. an oscillation observer module to detect and suppress
+oscillations introduced in the force-feedback control
+loop, and
+1https://www.xprize.org/prizes/avatar
+Preprint submitted to Robotics and Autonomous Systems
+January 3, 2023
+arXiv:2301.00764v1  [cs.RO]  2 Jan 2023
+
+Figure 1: Our bimanual haptic telemanipulation system: Human operator in operator station (right) inspecting an object during the ANA
+Avatar XPRIZE Competition semifinals through a remote anthropomorphic avatar robot (left).
+5. subsystem evaluation in lab experiments, a user study,
+as well as our participation at the ANA Avatar XPRIZE
+competition semifinals.
+2. Related Work
+Teleoperation is a widely investigated research area. A
+leading device (in our context called the Operator Station,
+see Section 3), often with haptic feedback is used to control
+a following device (Avatar Robot) in a remote location.
+The DARPA Robotics Challenge (DRC) 2015 [1] required
+the development of mobile telemanipulation systems. Sev-
+eral research groups, such as DRC-HUBO [3], CHIMP [4],
+RoboSimian [5], and our own entry Momaro [6] presented
+teleoperation systems with impressive manipulation capa-
+bilities. The focus was on completing as many manipu-
+lation and locomotion tasks as possible using a team of
+trained operators.
+Thus, some hardware and software
+components were highly specialized towards solving pre-
+defined tasks. In addition, the robots were not required to
+communicate or interact with other humans in the remote
+location and thus did not feature respective capabilities.
+In contrast, our developed avatar solution was designed for
+interaction with humans in the remote location and the
+operator interface is designed to give intuitive control over
+the robot to a single, possibly untrained operator. Pres-
+ence of the operator in the remote location is prioritized
+over specific task solution skills.
+Passivity control constitutes a large research field in
+the context of teleoperation. Uncertainties of the opera-
+tor’s input dynamics, as well as the remote environment
+are factors which introduce potential instability in control
+loops. Many different passivity control methods tackle the
+stability problem, e.g. [7, 8, 9, 10, 11, 12]. These control
+schemes use the concept of passivity which is a sufficient
+condition to obtain a stable control system. Control sys-
+tems are considered passive if and only if the energy flow-
+ing into the system exceeds the energy flowing out at any
+time. Conveniently, if all subsystems are passive the entire
+system is guaranteed to be passive as well.
+Ryu and Preusche [11] use a time-domain passivity con-
+trol approach to ensure stable teleoperation, handling time
+delay of up to 120 ms. A passivity observer is used to mon-
+itor the energy transferred from the operator to the avatar
+system and vice versa. The passivity controller actively
+dampens the system to ensure passivity and thus stability
+of the system. Our approach uses a similar observer and
+dampening approach to ensure stable teleoperation control
+loops. One drawback of energy-based time-domain passiv-
+ity controllers is the occurrence of position drift, which is
+handled by [13, 14] and improved in a 1 DOF (degree of
+freedom) teleoperation setup by Coelho et al. [15]. The de-
+sign of our control architecture assures position drift-free
+teleoperation by commanding goal poses for each avatar
+hand. Small position derivations can occur due to motion
+execution. However, these are negligible small for our ap-
+plication (see Section 5.3.2). Overall, passivity control can
+suffer from distortion of the displayed environment [16].
+In presence of large communication time delays (in
+the order of seconds) between operator and operator sta-
+tion (e.g., in space teleoperation missions), predictive con-
+trol methods can improve task performance [17]. The re-
+mote robot tries to anticipate human control commands to
+complete partially transmitted instructions. Hauser [18]
+presents a prediction and planning method to assist tele-
+operation without assuming a finite set of candidate tasks.
+Nevertheless, this method is limited to a finite set of task
+types and needs a large number of training data to over-
+come this limitation.
+Our control approach is designed
+to limit the operator only within the underlying hardware
+2
+
+EINALS
+NG2021SEMIF
+TESTING20
+L
+FINALS
+MIFINALS
+JG2021
+STING 2021
+ANAD
+ANA
+RIZE
+NALS
+TAR
+G2021
+INALS
+NG2O2
+ANAD
+NAL
+IG20
+AN
+FINAconstrains, without constraining the task solution itself.
+Some recent approaches use teleoperation interfaces
+which only send commands to the robot without providing
+any force or haptic feedback to the operator [19, 20]. The
+advantage of such systems is clearly the low weight of the
+capture devices which hinder the operator only marginally.
+The downside is missing force or haptic feedback, espe-
+cially for tasks that cannot be solved with visual feedback
+alone, such as difficult peg-in-hole tasks.
+Other methods use custom-developed operator inter-
+faces including force and haptic feedback.
+Klamt et al.
+[21] developed a centaur-like platform for teleoperation in
+disaster response scenarios.
+The proposed teleoperation
+interfaces uses actuators located at the base of the de-
+vice and metallic tendons for torque transmission to the
+actuated joints. This approach benefits from light-weight
+moving parts with low inertia resulting in an easily back-
+drivable system. However, the used metallic tendons in-
+troduce some compliant behavior which need to be con-
+sidered. In our approach, we use off-the-shelf robotic arms
+with actuators located inside the joints.
+We utilize an
+external FT-Sensor to actively follow the operator’s arm
+movement. Guanyang et al. [22] use two haptic devices
+(3DoF rotational and 3DoF translational device) to con-
+trol a robot’s end-effector. Both devices can display con-
+tact forces with high stiffness due to mechanical design.
+However, limited workspace and thus the requirement of a
+motion mapping between operator station and controlled
+robot as well as using two devices to control full 6D motion
+are some drawbacks compared to our approach. In Abi-
+Farrajl et al. [23] the operator station is comparable to our
+approach, but the focus there is placed on haptic feedback
+for balance control of the bipedal humanoid avatar robot.
+Wearable haptic feedback devices [24] overcome the
+workspace constraints generated by stationary devices but
+are limited to displaying contact since they cannot create
+any force towards the operator. Other research projects
+focus on controlling a teleoperation system under time
+delays [25] or with two different kinematic chains on the
+avatar side [26].
+In contrast to the highlighted related research, our ap-
+proach focuses on off-the-shelf components which allow for
+easy replication and maintenance. Furthermore, the used
+robotic arms are replaceable with any other appropriate
+actuators with different kinematic chains, since the whole
+communication between the systems uses only the 6D end-
+effector pose.
+3. Hardware Setup
+The developed robotic teleoperation system consists
+of an operator station and an avatar robot, as shown in
+Fig. 1. The operator station allows the operator to con-
+trol the avatar from a remote location.
+It includes two
+robotic arms, hand exoskeleton, 3D Rudder foot paddle,
+and a head mounted display with additional sensors. The
+avatar robot is designed to interact with humans and in
+SenseGlove
+Panda Arms
+FT Sensors
+Schunk
+SVH Hand
+Operator
+Avatar
+Figure 2: Operator (left) and avatar (right) arm with used hardware
+components. For simplicity, only the right arm is shown. The axes
+depict the common hand frame which is used for control commands
+and feedback.
+human-made indoor environments and, thus, features an
+anthropomorphic upper body mounted on a mobile base.
+The operator station and the avatar robot are con-
+trolled with a standard desktop computer (Intel i9-9900K
+@ 3.60 GHz, NVidia RTX 2080) each. The communica-
+tion between these computers is achieved by a single Gi-
+gabit Ethernet connection. We successfully tested the sys-
+tem with artificial delay of up to 30 ms in both directions.
+Thus, our system allows operating the avatar from a dis-
+tant location. On the software side, the Robot Operating
+System (ROS) framework is used. Both, the operator sta-
+tion and the avatar robot run their own roscore. We use
+NimbRo Network2 for any communication between both
+roscores. The hardware design of the operator station and
+avatar robot is described in the following.
+3.1. Avatar Robot
+The avatar robot’s anthropomorphic upper body mim-
+ics the human arm configuration using two 7 DoF Franka
+Emika Panda arms, mounted in slightly V-shaped angle.
+The shoulder height of 110 cm above the floor allows con-
+venient manipulation of objects on a table, as well inter-
+action with both sitting and standing persons. The shoul-
+der width of under 90 cm enables easy navigation through
+standard doors.
+The Panda arms have a sufficient payload of 3 kg and
+a maximal reach of 855 mm. The extra degree of freedom
+gives some flexibility in the elbow position. While the arm
+measures joint torques in each arm joint, we mounted addi-
+tional OnRobot HEX-E 6-Axis force/torque sensors at the
+wrists for more accurate force and torque measurements
+close to the robotic hands, since this is the default loca-
+tion of contact with the robot’s environment (see Fig. 2).
+The avatar robot is equipped with two anthropomorphic
+hands. A 20 DoF Schunk SVH hand is mounted on the
+right side. The nine actuated DoF provide very dexterous
+manipulation capabilities. The left arm features a 5 DoF
+2https://github.com/AIS-Bonn/nimbro_network
+3
+
+Franka
+Arm
+FT
+Sensor
+Sense
+Glove
+Arm
+Controller
+Hand
+Controller
+6D pose
+7DoF torque
+Avatar
+Model
+6D forces/torques
+7D torque
+Finger pos.
+Finger forces
+Arm
+Controller
+Hand
+Controller
+Franka
+Arm
+FT
+Sensor
+Schunk
+Hand
+Finger cmds
+Motor currents
+Hand command
+Hand feedback
+6D pose
+7DoF positions
+7DoF torques
+7DoF positions
+6D forces/torques
+Operator Station
+Avatar Robot
+Figure 3: Control System overview. For simplicity, only the right side is depicted. The left side is controlled similarly, besides a different
+Hand Controller for the different Schunk hands.
+Schunk SIH hand for simpler but more force-requiring ma-
+nipulation tasks. Both hand types thus complement each
+other.
+The avatar’s head is equipped with two RGB cameras,
+a microphone, and a small screen displaying the animated
+face of the operator [27]. It is attached to the upper body
+using a 6 DoF UFactory xArm for free head movement. In
+addition, two wide-angle RGB cameras are capturing the
+robot’s vicinity for situation awareness during locomotion.
+Further details on the VR remote visualization system are
+provided in [28]. The anthropomorphic upper body has
+been mounted on a movable base, which allows omnidirec-
+tional movement.
+3.2. Operator Station
+The operator controls the avatar through the Opera-
+tor Station from a comfortable sitting pose. The human
+hand movement is captured with a similar setup as al-
+ready described for the avatar robot: Two Panda arms
+are equipped with an OnRobot HEX-E force/torque sen-
+sor and connected to the operator hand using a SenseGlove
+haptic interaction device. The Panda arms thus serve dual
+purposes: They provide precise 6D human hand pose mea-
+surements for avatar control, as well as the possibility to
+induce force feedback measured by the Avatar onto the hu-
+man wrists. The operator-side force/torque sensor is used
+to measure the slightest operator hand movement to assist
+the operator in moving their arm, reducing the felt mass
+and friction to a minimum.
+The SenseGlove haptic interaction device features 20 DoF
+finger joint position measurements (four per finger) and a
+1 DoF haptic feedback channel per finger (i.e., when acti-
+vated the human feels resistance, which prevents further
+finger closing movement).
+For visual and audio communication, the operator is
+wearing a head mounted display equipped with eye track-
+ers, audio headset, and a camera viewing the lower face
+part (for more details see [29] and [30]). The avatar loco-
+motion can be controlled using a 3D Rudder foot paddle
+device.
+The Panda arms feature built-in safety measures and
+will stop immediately if force, torque, or velocity limits
+are exceeded. This ensures safe human-robot interactions
+both on the operator and the avatar side.
+4. Force Feedback Controller
+The control architecture for the force feedback tele-
+operation system consists of two arm and two hand con-
+trollers (one for each the operator and the avatar side). For
+the right and the left arm, each controller pair is running
+separately. The hand controller for the right and left hand
+are slightly different since different robotic hands are used.
+An overview of the control architecture is shown in Fig. 3.
+The arm controllers run with an update rate of 1 kHz and
+the force-torque sensor measurements are captured with
+500 Hz. The force-torque measurements are smoothed us-
+ing a sensor-sided low-pass filter with a cut-of frequency
+of 15 Hz.
+Since the robot arms are attached from outside to the
+operator’s wrists (see Fig. 3), the kinematic chains of avatar
+and operator station differ, and thus, a joint-by-joint map-
+ping of the operator and avatar arm is not possible. Con-
+sequently, the developed control concept does not rely on
+similar kinematic chains. Instead, a common control frame
+is defined in the middle of the palm of both the human and
+robotic hands, i.e., all necessary command and feedback
+data are transformed such that they refer to this frame
+before being transmitted. The controllers for the opera-
+tor and avatar arms and both hands are described in the
+following subsections.
+4.1. Operator Arm Controller
+The operator arm controller commands joint torques
+to the Panda arm and reads the current operator hand
+4
+
+robot
+SCHUNKOFigure 4: Unintended lower arm contact: Typical situation in which
+the avatar’s lower arm establishes contact with the environment,
+which cannot be measured by the force-torque sensor. Panda arm
+torques are used to provide the operator with appropriate force feed-
+back.
+tp
+1
+2 tp
+0
+0
+0.5
+1
+dp [rad]
+α
+Figure 5:
+Operator arm torque command (see Eq. (1)) is scaled
+using α to reduce oscillations when getting close to joint position
+or velocity limits. The scalar decreases linearly if the distance to a
+joint position limit dp exceeds the threshold tp. Velocity limits are
+handled analogously.
+pose to generate the commanded hand pose sent to the
+avatar robot.
+The goal is to generate a weightless feel-
+ing for the operator while moving the arm—if no force
+feedback is displayed. Even though the Panda arm has a
+convenient teach-mode using the internal gravity compen-
+sation when zero torques are commanded, the weightless
+feeling can be further improved by using precise external
+force-torque measurements.
+Any contact established by
+the avatar robot with its hands and lower arms is hap-
+tically displayed to the operator. Since the teleoperation
+system has no information about the operator’s intention,
+contact should not be avoided but displayed to the opera-
+tor to keep the human in control of the situation.
+For simplicity, just one arm is mentioned in the follow-
+ing, since the left and right arms are controlled equally.
+4.1.1. Torque Controller
+Let us denote with τo ∈ R7 the commanded joint torques
+for a particular time step. Then
+τo = ατcmd + βτf + τlo + τla + τno + τco
+(1)
+describes the used torque components (command, force
+feedback, operator limit avoidance, avatar limit avoidance,
+null-space, and Coriolis) which will be explained in the fol-
+lowing. Note that the gravity compensation is not consid-
+ered here, since it is done by the Franka Control Interface
+(FCI) itself.
+The commanded joint torques τcmd to move the Panda
+arm based on the force/torque sensor measurements and
+are defined as
+τcmd = JT F
+(2)
+with J being the body Jacobian relative to the hand frame
+and F ∈ R6 denoting the measured 3D forces and 3D
+torques. The scalars α ∈ R7 are computed by the predic-
+tive limit avoidance module (see Section 4.1.2). Note that
+F has to be corrected taking sensor bias and attached end-
+effector weight into account, as well as transformed into
+the common hand control frame (see Section 4.3).
+The term τf denotes the force feedback induced by
+the avatar-side force-torque sensor and Panda arm torque
+measurements. The scalar β is computed by the oscillation
+observer module (see Section 4.1.3) to prevent possible os-
+cillations in the feedback loop. The force-torque measure-
+ments are used as the primary feedback source. They are
+already bias-corrected and correctly transformed, there-
+fore Eq. (2) can be directly applied analogously to compute
+the induced joint torques τsensor.
+In some situations, especially when performing manip-
+ulation tasks on a table, the avatar establishes contact
+between the lower arm and the environment (for example
+the table, see Fig. 4), which are not visible from the opera-
+tor’s view pose. Since the force-torque sensor and hand are
+above the table, this type of contact cannot be measured
+by the force-torque sensor. Here, we must use the joint
+torque measurements of the Panda arm to give the opera-
+tor feedback about the contact. The Franka API provides
+estimated Cartesian forces fpanda at the end-effector. We
+use
+fdiff = ˆfpanda − ˆfsensor
+(3)
+to calculate the forces in the end-effector frame which
+cannot be measured using the force-torque sensor, where
+ˆfpanda and ˆfsensor are the respective low-pass filtered end-
+effector forces. Finally, we transform the calculated forces
+into the wrist frame and compute
+τf = τsensor + τdiff,
+(4)
+where τdiff ∈ R6 is fdiff ∈ R3 extended with t0 ∈ R3,
+since we ignore torque measurements here.
+4.1.2. Predictive Limit Avoidance
+Humans can achieve high speeds moving their arm,
+which can exceed the Panda joint velocity limits (up to
+150◦/s).
+To prevent the operator from exceeding joint
+position or velocity limits of the Panda arm, the term
+τlo ∈ R7 is introduced to apply torques pushing the arm
+away from those limits. For a single joint i, the torque to
+avoid its position limit is defined as
+5
+
+0
+1
+2
+3
+4
+5
+6
+7
+0
+1
+2
+3
+Initial
+Contact
+Time [s]
+Amplitude [N]
+x-Axis
+y-Axis
+z-Axis
+Figure 6: Oscillation observer: Frequency analysis (left) of force feedback measurements from the avatar robot’s right arm while placing a
+vase on a table (right). The initial contact between vase and table is marked. The graph shows the amplitude of the 4th frequency (ca.
+5.7 Hz) computed using a sliding DFT for each axis over time.
+Figure 7: Arm workspace evaluation. Left: Initial arm setup, simi-
+lar to the avatar side. Right: Optimized mounting pose. Turquoise
+(reachable) and red (not reachable) arrows depict the captured hu-
+man left-hand poses. The coordinate axes depict the operator sitting
+pose.
+τ i
+lo−position =
+�
+γp( 1
+di
+p − 1
+tp ),
+di
+p < tp
+0,
+else
+(5)
+with γp being a constant scalar, di
+p being the distance for
+joint i to its closer position limit, and tp = 10◦ being a
+threshold how close a joint must be at a limit to activate
+this behavior. τ i
+lo−velocity is calculated analogously with
+tv = 40◦/sec. Together, τlo is defined as
+τlo = τlo−position + τlo−velocity.
+(6)
+The torques τlo exhibit hyperbolical growth when getting
+closer to respective limits. Since the operator-side force-
+torque sensor will measure the generated limit avoidance
+torques, the arm can end up oscillating, especially being
+close to one or multiple position limits. Thus, the torques
+τcmd, which are influenced by the force-torque sensor, are
+scaled per joint by α, which is defined as
+α = max(0, min(1, 2 min(dp
+tp
+, dv
+tv
+) − 1)).
+(7)
+The scalar α is designed to decrease linearly and reach zero
+when the limit is approached halfway after activating the
+limit avoidance (see Fig. 5). This reduces the commanded
+torques τcmd enough when approaching a position or ve-
+locity limit and prevents the oscillation.
+As already mentioned, the operator station and avatar
+robot have different kinematic arm chains. Therefore, avoid-
+ing position and velocity limits on the operator side does
+not guarantee limit avoidance on the avatar side. Calculat-
+ing the joint torques preventing joint limits on the avatar
+robot in a similar way is not beneficial, since the feedback
+information would arrive with high latency (mainly be-
+cause of the delay generated by motion execution on the
+avatar side). To overcome this issue, we use a model of
+the avatar inside the operator arm controller to predict the
+avatar arm movement for the next time step and calculate
+the needed joint torques to prevent joint limit violations
+in advance.
+The current operator hand pose is used as the desired
+goal pose for the avatar arm in the common gripper frame.
+We estimate the avatar arm joint configuration reaching
+this goal pose using inverse kinematics (IK). The latest re-
+ceived avatar arm joint positions are used to initialize the
+IK solver. The current joint velocities are approximated by
+a low-pass filtered version of the joint position first deriva-
+tives. Having estimated the joint positions and velocities,
+we can apply the same avoidance strategy as described
+above (see Eq. (6)). Finally, the resulting joint torques
+can be transformed into the common 6D hand frame us-
+ing the pseudoinverse of the Jacobian transpose (JT
+A)+ and
+back to joint torques for the operator arm with JT
+O. This
+results in
+τla = JT
+O(JT
+A)+τla−model.
+(8)
+The remaining two torque components from Eq. (1)
+are τno and τco. The term τno is a null-space optimiza-
+tion term which pulls the elbow towards a defined conve-
+nient pose in the null-space of the Jacobian. The result
+is a more human-like elbow pose to maximize the oper-
+6
+
+ator workspace by pushing the arm away from singulari-
+ties. The last torque component is the Coriolis term τco
+obtained by the Panda model.
+4.1.3. Oscillation Observer
+Our telemanipulation control system as described above
+is designed as simple as possible, but yet powerful to give
+the operator control over the robot along with necessary
+force feedback. The downside is, that our controller can-
+not promise any stability, resulting in oscillations when
+certain contact forces are measured. We address this is-
+sue by observing the force feedback channel, detecting any
+critical oscillations, and reducing the feedback introduced
+into the system.
+The sliding Discrete Fourier-Transformation (DFT) [31,
+32] method is used to analyze the force feedback channel
+per axis. Since we did not measure isolated oscillations in
+the torque channels, analyzing the 3D forces is sufficient.
+The force measurements are sampled with around 1 kHz.
+We experimentally investigated the oscillation frequency
+to obtain suitable DFT parameters and the observed fre-
+quency band. In the experiments, we provoked vibrations
+in the system by placing a vase harshly on a table. The
+used sliding DFT has resolution of 512 and uses the Han-
+ning window to minimize spectral leakage.
+We observe
+correlation between the generated oscillation and the am-
+plitude of the 4th frequency (ca. 5.7 Hz), is depicted in
+Fig. 6.
+In each time step, the force measurements of all three
+axes are analyzed using separate sliding DFTs. Next, the
+combined result is computed using the Euclidean norm
+over all axes amplitudes. The resulting value v is clamped
+using experimentally obtained parameters (min = 163 and
+max = 500) and scaled to be within the interval [0, 1].
+Finally, the scalar β = 1 − v is used to scale the force
+feedback provided to the operator (see Section 4.1.1). The
+change rate of β is limited to gradually remove and re-
+store the force feedback, s.t. the observer needs 0.8 s to
+fully remove the feedback in case of detected oscillations
+and 1.7 s to reach the normal force feedback control sta-
+tus again. This gentle oscillation elimination behavior is
+as little noticeable for the operator as possible. Note that
+Fig. 6 shows the oscillation without feedback reduction.
+The computation of β and the effect oscillation damping
+are evaluated in Section 5.1.
+The avatar arm controller commands the Panda arm
+of the avatar robot to follow the commanded 6D pose by
+sending joint torques τa ∈ R7 to the Franka Control Inter-
+face. The commanded torque is defined as
+τa = τcmd + τinit + τna + τca,
+(9)
+where τcmd ∈ R7 and τinit ∈ R7 are calculated to reach
+the goal pose during operation and initialization, respec-
+tively (see below). The components τna and τca are the
+null-space optimization and Coriolis terms similar to τno
+and τco, as described in Section 4.1. The convenient elbow
+poses used for the null-space optimization is defined such
+that the elbows are slightly stretched out. This generates
+a more human-like arm configuration and keeps the arm
+away from singularities in the elbow joints. Other singular-
+ities do not occur due to the nature of the arm kinematic
+in the usable workspace and other joint position limits.
+Therefore, no special singularity handling is needed here.
+The goal torques τcmd and τinit are generated using
+a Cartesian impedance controller that emulates a spring–
+damper system. Its equilibrium point is the 6D goal pose
+commanded by the operator station in the common hand
+frame:
+τcmd = JT (−S∆p − D(J ˙q)),
+(10)
+where J denotes the zero Jacobian, S ∈ R6×6 and D ∈
+R6×6 denote the stiffness and damping matrix, ∆p ∈ R6 is
+the error in translation and rotation between the current
+and goal end-effector pose, and ˙q ∈ R7 denotes the current
+joint velocities. τinit is only used for a safe initialization
+procedure (see below) and generated similarly using the
+current end-effector pose. The stiffness and damping pa-
+rameters symbols and their values are empirically tuned
+to achieve some compliance while still closely following the
+operator command.
+When no goal pose command is received, the controller
+keeps commanding the current arm pose to remain in a safe
+state. This happens when the operator station is not ac-
+tive or if a communication breakdown occurs (no operator
+command within the last 100 ms). After receiving a com-
+mand, the controller performs an initialization procedure
+which fades linearly between the current and the new re-
+ceived goal pose. This prevents the robot from generating
+high torques to suddenly reach the new, possibly distant
+pose. This initialization process takes about 3 s.
+The Panda arm stops immediately when excessive forces
+are measured, for example when there is unintended con-
+tact that exceeds force/torque thresholds. This feature is
+necessary to operate in a safe way. After notification of
+the human operator, the avatar arm controller can restart
+the arm automatically. After performing the initialization
+procedure, normal teleoperation can be resumed.
+4.2. Hand Control
+The operator finger movements are captured using two
+SenseGlove haptic interaction devices. Four separate fin-
+ger joint measurements are provided per finger. Since the
+Schunk SVH and SIH robotic hands on the avatar have
+nine and five actuated joints, respectively, only the cor-
+responding joint measurements are selected and linearly
+mapped to the avatar hands.
+While this mapping does
+not precisely replicate hand postures – this is impossible
+anyways due to the different kinematic structure – it gives
+the operator full control over all hand DoFs.
+Both hands provide feedback in the form of motor cur-
+rents, which is used to provide per-finger haptic feedback
+to the operator. The SenseGlove brake system is switched
+on or off depending on a pre-defined current threshold.
+7
+
+0
+0.5
+1
+1.5
+2
+2.5
+−1.05
+−1
+−0.95
+∆t
+Time [s]
+q1 [rad]
+Figure 8: Predictive avatar model: Measured joint position for the
+first joint of the right avatar arm during a grasping motion (green)
+and predicted joint position for predictive limit avoidance (blue).
+Both measurements are captured on the operator side. Communica-
+tion between both systems and motion execution generate a delay of
+up to 200 ms (∆t), which is compensated by the predictive model.
+4.3. Force-Torque Sensor Calibration
+Different end-effectors (SenseGloves, Schunk SIH, Schunk
+SVH hand, and corresponding 3D printed mounting adapters)
+are mounted on each of the four involved force-torque sen-
+sors. In addition, sensor bias results in barely usable raw
+sensor data. Thus, each sensor is calibrated separately to
+compensate these effects. To this end, 20 data samples
+from different sensor poses are collected. Each sample in-
+cludes the gravity vector in the sensor frame and the mean
+of 100 sensor measurements from a static pose. A standard
+least squares solver [33] is used to estimate the force-torque
+sensor parameters, i.e., the force and torque bias and the
+mass and center of mass of all attached components. The
+same parameters including the additional mass and center
+of mass transformation resulting by the force-torque sensor
+itself is used to configure the built-in gravity compensation
+of the Panda arms. The calibration is performed once after
+hardware changes at the end-effectors or if the bias drift is
+too large. This method does not compensate for bias drift
+during usage, but is sufficient for our application.
+5. Evaluation
+In addition to our participation at the ANA Avatar
+XPRIZE Competition semifinals, we performed multiple
+experiments along with a small user study to evaluate the
+developed teleoperation system in our lab environment.
+5.1. Quantitative Experiments
+In a first experiment, we evaluated the operator arm
+workspace. 2,959 different 6D left hand poses were cap-
+tured from a sitting person performing typical arm mo-
+tions with a VR tracker on their wrist.
+In addition to
+hand poses with a fully extended arm, most of the poses
+are directly in front of the person, likely to be performed
+during manipulation tasks. First, the initial arm mount-
+ing pose (motivated by the avatar configuration) of the op-
+erator arm was evaluated. Each captured hand pose was
+marked as reachable if an inverse kinematic solution for the
+0
+50
+100
+150
+200
+250
+−10
+0
+10
+Ours
+Time [s]
+Force [N]
+0
+50
+100
+150
+200
+250
+−10
+0
+10
+Panda
+Time [s]
+Force [N]
+Figure 9: Operator arm movement: Force in z-direction (in the di-
+rection of the human palm) needed to move the arm in the same
+repetitive motion with our operator arm controller running (top)
+and using only the Panda built-in gravity compensation (bottom).
+Table 1: Operator arm workspace analysis
+Mounting Pose
+Reached
+Missed
+Reached [%]
+Initial
+1,795
+1,164
+60.6 %
+Optimized
+2,848
+111
+96.3 %
+arm was found. In a second step, different arm mounting
+poses were sampled to find an optimal pose, maximizing
+the number of reachable hand poses (see Fig. 7). Table 1
+reports quantitative results.
+The resulting arm mount-
+ing pose drastically increases the overlap (from 60.6% to
+96.3%) between the human operator’s and the avatar’s
+arm workspace, but requires a more complicated mounting
+setup.
+In a second experiment, we evaluated the predictive
+limit avoidance module. Avatar arm position and veloc-
+ity limits are haptically displayed via joint forces to the
+operator.
+Since measured joint positions and velocities
+are afflicted with latency generated by network commu-
+nication (<1 ms) and motion execution using the Carte-
+sian impedance controller, which can reach up to 200 ms
+(see Section 4.1.1), the operator control predicts the avatar
+arm joint configuration. Fig. 8 shows the measured and
+predicted joint position of the first right arm joint. The
+prediction compensates the delays, which allows for instan-
+taneous feedback of the avatar arm limits to the operator.
+In a third experiment, we investigated the forces and
+torques required to move the operator station arm, since
+this directly affects operator fatigue.
+We measured the
+forces and torques applied to the arm by reading the force-
+torque sensor measurements.
+The arm was moved in a
+comparable manner once with only the Panda gravity com-
+pensation enabled (i.e., τcmd = 0 see Eq. (1)) and a second
+8
+
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0
+0.5
+1
+1.5
+Initial
+Contact
+Active Observer
+Time [s]
+Amplitude [N]
+x-Axis
+y-Axis
+z-Axis
+(a)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+0
+0.5
+1
+1.5
+2
+2.5
+Initial
+Contact
+Inactive Observer
+Time [s]
+Amplitude [N]
+x-Axis
+y-Axis
+z-Axis
+(b)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+−1
+0
+1
+2
+Time [s]
+Joint Torque [Nm]
+0
+0.5
+1
+Initial
+Contact
+Active Observer
+β [0, 1]
+τ7
+β
+(c)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+−1
+0
+1
+2
+3
+Time [s]
+Joint Torque [Nm]
+0
+0.5
+1
+Initial
+Contact
+Inactive Observer
+β [0, 1]
+τ7
+β
+(d)
+Figure 10: Oscillation observer module. Placing a vase onto a ta-
+ble results in the depicted amplitude response for 5.7 Hz for each
+Cartesian axis ((a) and (b)). We performed the experiment twice
+with comparable executions. First, the oscillation observer was ac-
+tive ((a) and (c)). In the second execution, the observer was inactive
+((b) and (d)). The corresponding joint torques commanded to the
+operator Panda arm are plotted exemplary for the 7th joint (c) and
+(d). The force feedback is scaled with β (c) which is computed from
+the oscillation observer. Note that β shown in (d) was computed but
+not used during execution.
+time with our arm force controller running. In Fig. 9, the
+forces in the direction of one exemplary axis are shown.
+The results demonstrate the advantage of using an exter-
+nal force-torque sensor to generate a more unencumbered
+feeling for the operator while using the system.
+In the last experiment, we analyzed the oscillation ob-
+server module (see Section 4.1.3), which observes the avatar
+force feedback channel to detect and prevent oscillations in
+the closed-loop controller. We used the avatar system to
+place a metal vase harshly on a table (see Fig. 6). The exe-
+cuted joint torques on the operator side along with the fre-
+quency responses of the force-feedback measurements are
+shown in Fig. 10. Although the oscillations are not elim-
+inated immediately (c), reducing the feedback gain yields
+the expected oscillation suppression. The oscillation is de-
+tected by analyzing the frequency response shown in (a)
+and (b). The gain β is reduced with the maximum allowed
+rate to a value of 0.1 within 0.8 s. Next, the operator does
+not feel the contact forces as strong, the oscillation stops
+(shown in the exemplary torque command for the 7th oper-
+ator arm joint (c)), and the gain is increased again, which
+brings back the force feedback for the operator. In conclu-
+sion, the oscillation observer module is a necessary part
+of our simple force-feedback control loop to ensure safe
+operation.
+5.2. User Study
+Our goal was to create an immersive and intuitive feel-
+ing for operation at remote locations using our system.
+Since humans have their very own preferences and sub-
+jective feelings of how good or intuitive certain control
+mechanisms perform, we carried out a user study with
+untrained operators, comparing different telemanipulation
+approaches. Due to the COVID-19 pandemic, we were lim-
+ited to immediate colleagues as subjects, which severely
+constrained the scope of our study. Although all partici-
+pants had a rough idea of the system, they controlled it
+for the first time during this study.
+A total of five participants were asked to perform a bi-
+manual peg-in-hole manipulation task. First, two different
+objects had to be grasped: a small aluminum bar and a 3D
+printed part with a hole. Afterwards, the bar should be
+inserted into the hole (see Fig. 11). The avatar robot was
+already placed in front of a table and both objects were
+within the avatar’s workspace. Participants controlled the
+robot using the operator station located within the same
+room. Only the HMD with the avatar’s perspective was
+used for visual feedback. The task was challenging due to
+very little friction between the finger and objects and tight
+tolerances, which required precise insertion alignment.
+Each participant performed the task three times with
+the following control modes:
+1. Operator station with force feedback enabled,
+2. Operator station with force feedback disabled, and
+3. VR controllers.
+9
+
+Figure 11: User study with untrained operator: Both objects had to
+be grasped and the bar had to be inserted into the hole.
+Table 2: User study success rates and timings.
+Telemanipulation mode
+Success
+Completion time [s]
+Mean
+StdDev
+1) Exoskeleton with feedback
+4/5
+119.0
+117.1
+2) Exoskeleton w/o feedback
+5/5
+123.0
+88.4
+3) VR controllers
+3/5
+126.3
+25.8
+In the first control mode, all system components were
+active as described in this article. Any force and haptic
+feedback were disabled for the second control mode, i.e.
+the operator had to rely on visual feedback only. In the
+third control mode two HTC Vive VR controllers were
+used as input devices. As long as the trigger button was
+pressed, the corresponding avatar arm followed the con-
+troller movement. A different button was programmed to
+toggle between a defined closed and open hand pose.
+A maximum of 5 min were granted to solve the task be-
+fore it was marked as a failure. An object dropped outside
+the reachable workspace resulted in a failed trial. Objects
+dropped onto the table within the workspace of the avatar
+could be grasped again with no penalty. The participants
+were allowed to test each control mode about 1 min before
+starting the measured test.
+Table 2 reports the quantitative results of the user
+study. The time needed to successfully solve the task is
+quite similar over the different telemanipulation modes.
+From these experiments, we realized that the completion
+time was highly influenced by external factors, such as
+losing the object due to not enough finger friction or dif-
+ferent grasping and object handling solutions, which are
+unrelated to the used operator interface. In addition, hu-
+mans can easily compensate missing force feedback using
+visual feedback. All three unsuccessful trials failed due to
+reaching the maximum experiment time of 5 min. To gen-
+erate a more meaningful statement based on performance
+scores, more test samples are needed.
+In addition to these quantitative measurements, we
+asked the participants to answer a short questionnaire
+about each telemanipulation mode with answers from the
+1-7 Likert scale (see Fig. 12). The results show that es-
+pecially the feeling of handling the objects and intuitive
+finger control was subjectively much better using the Op-
+erator Station.
+Enabling the force and haptic feedback
+gives the highest advantage when picking up the objects
+Table 3: Avatar Arm safety stops
+Scenario 1
+Scenario 2
+Scenario 3
+Day 1
+Exceeded
+Exceeded
+None
+torque limits
+vel. limits
+Day 2
+None
+None
+Software failure
+from the table. This can be explained by the additional
+feedback indicating contact between the hand and the ta-
+ble which cannot be perceived visually due to occlusions.
+All participants reported to feel safe and comfortable us-
+ing the system. Although the experiment time was limited,
+this suggests non-excessive cognitive load on the operator.
+Overall, the user study showed that our developed sys-
+tem is intuitive to use for untrained operators. Even though
+the force and haptic feedback did not increase the success
+rate of solving the task, it increases the immersive feeling
+as shown by the questionnaire.
+5.3. ANA Avatar XPRIZE Competition
+Our avatar system was evaluated by independent judges
+during the ANA Avatar XPRIZE competition semifinals
+over two days3.
+At each competition day, a first judge
+acted as the operator who performed 18 tasks in three
+predefined scenarios together with the second judge act-
+ing as the so called “recipient”. The recipient was sitting
+with the avatar robot at a table in a different room about
+100 m away from the operator control room. Communi-
+cation of any kind between both judges was only possible
+through the avatar system. Both judges were trained to
+get comfortable with our system during the first hour of a
+trial. In the second hour, the judges had to solve the tasks
+without any instructions or support from our team. Thus,
+the judges were no experts but slightly more trained com-
+pared to the completely untrained operators in our user
+study (see Section 5.2). Fig. 13 shows the three scenar-
+ios: Solving a jigsaw puzzle, celebrating a business deal,
+and exploring an artifact from a historical exhibition with
+the robot’s senses. The enabled force feedback (Control
+Mode 1, see Section 5.2) enabled the operator to feel some
+texture of the artifact. The judges evaluated the avatar
+system with a major focus on the ability to convey human
+senses, actions, and presence in the remote location in real
+time. At both competition days, the same three scenar-
+ios were tested by different judges. The better score per
+scenario (max. 30 points per scenario) counted towards
+the final score. Additional 10 points were given based on
+a video submitted prior to the semifinal showing the sys-
+tem in action in our lab4. Our team NimbRo archived an
+almost perfect score of 99 out of 100 points, which placed
+us first in the semifinal.
+3Video material about the competition can be found here: http:
+//ais.uni-bonn.de/nimbro/AVATAR/
+4https://www.youtube.com/watch?v=yGwJIDBMolk
+10
+
+1
+2
+3
+4
+5
+6
+7
+Did you feel safe and comfortable?
+Did you feel like you were handling the objects directly?
+Was it easy to control the robot?
+Was it intuitive to control the arms?
+Was it intuitive to control the fingers?
+Did you find and recognize the objects?
+Was it easy to grasp the objects?
+Was it easy to fit the objects together?
+VR
+Without FF
+With FF
+better
+Figure 12: Qualitative results of our user questionnaire. We show the median, lower and upper quartile (includes interquartile range), lower
+and upper fence, outliers (marked with •) as well as the average value (marked with ×), for each aspect as recorded in our questionnaire.
+Figure 13: ANA Avatar XPRIZE Semifinal Scenarios (left) and the
+used objects (right).
+The three scenarios were: Solving a jigsaw
+puzzle (top), celebrating a business deal (middle), and exploring an
+artifact (bottom).
+We analyzed the recorded data during the official semi-
+final test runs to obtain more insight on the technical per-
+formance of our system. In the following, we report some
+results.
+5.3.1. Safety System
+One important aspect of our avatar robot is to operate
+safely next to and in cooperation with humans. Therefore,
+the avatar arm controller stops the Panda arm immedi-
+ately when unexpected high forces/torques are measured
+(see Section 4.1.1).
+This behavior was activated three
+times during the performance of the six official scenarios
+(see Table 3).
+At Day 1, both arm stops were triggered by exceeded
+torque and velocity limits. In the first case, operator and
+recipient performed a powerful ’high five’ gesture, which
+resulted in a rapid increase of torque applied to the robot,
+exceeding the joint torque limits.
+The predictive limit
+avoidance module was not able to prevent the operator
+executing such high contact forces, since the operator arm
+torque limits were reached as well. In the second scenario
+on Day 1, the operator waved to the recipient. The dif-
+ferent kinematic chains of the operator and avatar arm
+demand high joint accelerations and speeds on the avatar
+side for relative low operator arm speeds and accelerations.
+The limit predictive module tried to slow down the oper-
+ator waving speed, but humans can overcome these ap-
+plied feedback forces to always give the operator control
+for safety reasons. Here, the operator learned immediately
+continuing waving in a slightly slower and less acceleration-
+needing manner. At the second competition day, the right
+avatar arm stopped once while not actively being moved.
+After some investigation, we found a software failure which
+led to a very unlikely race condition. This was easily fixed
+after the competition.
+In all cases, the avatar and operator robot continued
+to work in a safe manner, which is necessary for human-
+robot interactions. Our intuitive feedback to the operator
+11
+
+NimbRo Avatar
+Avatar XPRIZE Semifinals
+XPRIZE
+ANA
+AVATAR
+SEMIFINALS
+TESTING202
+TINALO21
+ANAD
+ANAD
+ANA
+VOALONimbRo Avatar
+X
+Avatar XPRIZE Semifinals
+区
+XPRIZE
+ANA
+AVATAR
+SEMIFINALS
+TESTING2
+IAL021
+ANAD
+AR
+ANAD
+OTRNNimbRo Avatar
+X
+Avatar XPRIZE Semifinals
+XPRIZE
+ANA
+AVATAR
+SEMIFINALS
+TESTING2O
+口日
+FINAL021
+ANAD
+RTAR
+ANAL
+AR
+ANAC
+ORNFigure 14: Workspace analysis for all scenarios (solving a puzzle, celebrating a business deal, exploring an artifact) during the ANA Avatar
+XPRIZE Competition semifinal. Top: Hand positions of the avatar (blue) and operator (magenta) depicted in the common frame. Bottom:
+Operator VR perspective.
+0
+20
+40
+60
+80
+100
+6.4
+6.6
+6.8
+Temporal Offset [ms]
+Translation Error [mm]
+Figure 15: Avatar arm latency analysis. The graph shows the re-
+sulting mean translation error, comparing the current avatar hand
+position with an operator command shifted into the future.
+The
+mean is calculated over both arms and all six official competition
+scenarios. The minimum can be found at 44 ms, which is the esti-
+mated round-trip latency in our system — including network and
+motion execution.
+gave instantaneous situation awareness and allowed the
+operator to react and learn immediately to continue the
+ongoing tasks.
+5.3.2. Arm Controller Accuracy
+After optimizing the usable workspace for the opera-
+tor (see Section 5.1), Fig. 14 shows the used workspace
+for all three scenarios at the second competition day. The
+required workspaces during the competition tasks were rel-
+atively low, such that we had no issues with our system.
+We used the same data to analyze the spatial and tem-
+poral accuracy of our arm controller, i.e., how precise and
+fast the avatar arms follow the operator commands during
+an evaluation task.
+The measured operator and avatar hand positions were
+captured with 1000 Hz over the duration of the six official
+competition scenarios.
+We then calculated the transla-
+tion error between the corresponding target and goal posi-
+tion for each arm, resulting in a 6.6 mm mean translation
+error.
+In addition, we investigated the estimated delay
+in our arm controller including network communication,
+controller runtime, and motion execution. We compared
+the measured hand position of the avatar with the up to
+100 ms delayed operator’s arm control position and cal-
+culated the mean translation error as explained above.
+Fig. 15 shows the mean translation error over all com-
+petition tasks and both arms for a given temporal offset.
+The minimum translation error is archived using a tempo-
+ral shift of 44 ms, resulting in a mean translation error of
+12
+
+oetaatUg
+FrameRat
+Normal Cell Count
+Cel su
+Calor
+Line Stylt
+160: 160:164
+Apha
+Ran
+oflst
+0,; 0; 0
+/Status O1
+Syle
+Decay Tim
+Colo
+21:641
+/Statux:O1
+Style
+oint
+Alpha
+DecayTim
+Position Trarsformt
+YZ258; 255; 255
+DefaultLigh
+Cel su
+LineStyl
+Colot
+160: 160:164
+Apha
+oflst
+Ryle
+ColorTr
+Y
+Colo
+21:641
+/Statux:O1
+Style
+Sine (Pioels
+DecayTim
+Trarsfom
+Yoefaut Ugn
+tormal CellCoun
+Cel su
+Line Stylt
+160: 160:164
+Apha
+oflst
+0,; 0; 0
+Decay Tim
+Y
+Colo
+21:641
+/Statux:O1
+Style
+oint
+Alpha
+Position Trarsformt
+YNimbRo Avatar
+Avatar XPRIZE Semifinals
+XPRIZE
+ANA
+AVATAR
+SEMIFINALS
+TESTING2O2
+口口日
+TINALO21
+ANAD
+VRTAR
+ANAD
+ANAD
+ORNNimbRo Avatar
+X
+Avatar XPRIZE Semifinals
+区1
+XPRIZE
+ANA
+AVATAR
+SEMIFINALS
+TESTING2
+TI0A2021
+ANAD
+ANAD
+AR
+ANAD
+ORINimbRo Avatar
+Avatar XPRIZE Semifinals
+XPRIZE
+ANA
+AVATAR
+SEMIFINALS
+TESTING 20
+口日
+TI0AL021
+ANAL
+RIZAR
+ANA
+ANAD
+OTRNTable 4: Translation error [mm] at ANA Avatar XPRIZE semifinals.
+Day 1
+Day 2
+Scenario
+1
+2
+3
+1
+2
+3
+Left Arm
+6.4
+5.7
+7.9
+4.9
+3.8
+4.8
+44 ms shift1
+5.8
+5.3
+7.6
+4.6
+3.6
+4.5
+Right Arm
+13.2
+9.5
+6.4
+6.3
+5.5
+5.1
+44 ms shift1
+12.5
+9.2
+6.0
+6.3
+5.2
+4.8
+1 Delay between operator station and avatar (see Fig. 15)
+6.3 mm. The results with and without temporal shift for
+both arms and all tasks are reported in Table 4. Notice-
+able is the comparably large error for the right arm during
+Scenario 1 and 2 at the first competition day. This can be
+explained by the executed safety stops (see Section 5.3.1).
+It takes about two seconds to safely fade the avatar arm
+to the commanded pose. This results in possibly large po-
+sition errors. Overall, the observed errors are rather small
+and not noticeable for the operator and could potentially
+be decreased by an even more accurate calibration on a
+whole-system level. The estimated round-trip execution
+latency of 44 ms could be considered for future system im-
+provement.
+6. Discussion & Conclusion
+This work presented a bimanual telemanipulation sys-
+tem consisting of an exoskeleton-based operator control
+station and an anthropomorphic avatar robot. Both com-
+ponents communicate using our force and haptic feedback
+controller, which allows safe and intuitive teleoperation for
+both the operator and persons directly interacting with the
+avatar. The control method is agnostic to the kinematic
+parameters and uses only a common Cartesian hand frame
+for commands and feedback.
+Using the predictive limit
+avoidance avatar model, arm limits for both the operator
+and avatar side can be force-displayed to the operator with
+low latency. The oscillation observer and damper mod-
+ules detect and suppress oscillations in the feedback con-
+trol loop by reducing the force feedback gains temporarily.
+Additional force-torque sensors measurements are used to
+generate a weightless feeling for the operator while moving
+the arms without establishing contact on the avatar side.
+We evaluated the system using a user study with un-
+trained operators as well as in lab experiments. In addi-
+tion, the system performed very well at the ANA Avatar
+XPRIZE Competition Semifinals, scoring 99 out of 100
+points. This demonstrates the intuitiveness and reliability
+of our system and its control methods.
+Acknowledgments
+This work has been funded by the Deutsche Forschungs-
+gemeinschaft (DFG, German Research Foundation) under
+Germany’s Excellence Strategy,
+EXC-2070 - 390732324 - PhenoRob.
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+

8NAyT4oBgHgl3EQfc_et/content/tmp_files/2301.00294v1.pdf.txt

@@ -0,0 +1,1685 @@
+Modular Hamiltonian of the scalar field in the semi
+infinite line: dimensional reduction for spherically
+symmetric regions
+Marina Huerta∗ and Guido van der Velde†
+Centro Atómico Bariloche, 8400-S.C. de Bariloche, Río Negro, Argentina
+Abstract
+We focus our attention on the one dimensional scalar theories that result from dimen-
+sionally reducing the free scalar field theory in arbitrary d dimensions. As is well known,
+after integrating out the angular coordinates, the free scalar theory can be expressed as
+an infinite sum of theories living in the semi-infinite line, labeled by the angular modes
+{ℓ, ⃗m}. We show that their modular Hamiltonian in an interval attached to the origin is, in
+turn, the one obtained from the dimensional reduction of the modular Hamiltonian of the
+conformal parent theory in a sphere. Remarkably, this is a local expression in the energy
+density, as happens in the conformal case, although the resulting one-dimensional theories
+are clearly not conformal. We support this result by analyzing the symmetries of these
+theories, which turn out to be a portion of the original conformal group, and proving that
+the reduced modular Hamiltonian is in fact the operator generating the modular flow in
+the interval. By studying the spectrum of these modular Hamiltonians, we also provide an
+analytic expression for the associated entanglement entropy. Finally, extending the radial
+regularization scheme originally introduced by Srednicki, we sum over the angular modes
+to successfully recover the conformal anomaly in the entropy logarithmic coefficient in even
+dimensions, as well as the universal constant F term in d = 3.
+1
+Introduction: Modular flow and modular Hamiltonian
+The successful application of information theory tools to quantum field theory (QFT) along the
+last decades, has given place to the solid current consensus that these tools must be definitively
+incorporated into the usual QFT machinery. In this context, the study of quantities related
+to different information measures for quantum field theories gains relevance and with them, the
+study of states reduced to a region. These states are described by reduced (local) density matrices
+that live in the core of the definition of all the information measures referenced to spatial regions
+R. From the quantum field algebraic perspective [1], each region R is attached to the algebra
+∗e-mail: [email protected]
+†e-mail: [email protected]
+1
+arXiv:2301.00294v1  [hep-th]  31 Dec 2022
+
+of the degrees of freedom localized in R. The reduced state to a local algebra of operators in a
+region can be expressed, in presence of a cutoff, as a density matrix
+ρ = e−K
+tre−K ,
+(1)
+where the exponent K is the modular Hamiltonian operator. This convenient way of encoding
+the reduced state admits an interesting interpretation of the entanglement entropy as the ther-
+modynamic entropy of a system in equilibrium at temperature 1, but with respect to the modular
+Hamiltonian K. Moreover, there is a time notion associated to the state through the modular
+Hamiltonian, whose evolution is implemented by the unitary operator in the algebra
+U(τ) = ρiτ ∼ e−iτK .
+(2)
+The induced evolution of operators O(τ) = U(τ)OU(−τ) is called the modular flow. This is a
+purely quantum transformation, which becomes trivial in the classical limit.
+Historically, the earliest recognition of the structural importance of modular flows can be found
+in the algebraic formulation of QFT [2, 3] and more recently, in the framework of the study of
+different information measures and statistical properties of reduced states in QFT [4, 5, 6].
+The modular Hamiltonian is a fundamental constitutive part of the relative entropy and plays
+an essential role in the entropy bounds formulations and proof of several energy conditions
+[7, 8, 9, 10, 11, 12]. Besides, profiting that entanglement and relative entropy have well established
+geometric duals for holographic QFT [13, 14, 15], modular Hamiltonians have also been used to
+clarify localization properties of degrees of freedom in quantum gravity [16, 17, 18].
+Currently, our knowledge of the explicit form of modular Hamiltonians reduces mostly to
+some examples where the modular flow is local, and it is primarily determined by spacetime
+symmetries.
+This is the case for the Rindler wedge x1 > |t| in Minkowski space and any QFT. Choosing the
+causal region to be the half spatial plane x1 > 0 and t = 0 then, the rotational symmetry of the
+euclidean theory allows us to express the reduced density matrix corresponding to the vacuum
+state in terms of the energy density T00
+ρ = k e−2π
+�
+x1>0 dd−1x x1T00(x) .
+(3)
+The above expression manifestly reveals a non trivial connection between entanglement in
+vacuum and energy density. Moreover, in equation (3), the exponent corresponds to the modular
+Hamiltonian for half space which results to be an integral of a local operator. K is in fact 2π
+times the generator of boosts restricted to act only on the right Rindler wedge
+K = −2π
+�
+x1>0
+dd−1x x1T00(x) .
+(4)
+The modular flow ρiτ moves operators locally following the orbits of the one parameter group
+of boost transformations. On the other hand, it is interesting to note that from equation (3),
+the vacuum state in half space corresponds to a thermal state of inverse temperature 2π with
+respect to the boost operator. This is directly connected to the Unruh’s effect [19] according
+to which accelerated observers see the vacuum as a thermally excited state. For an observer
+following a trajectory given by a boost orbit, the state looks like a thermal state with respect
+to the proper time ˜τ. For these trajectories, the proper time and the boost parameter s are
+2
+
+proportional s = a˜τ with a the proper acceleration of the observer, constant along boost orbits.
+In turn, this implies there is a relation K = ˜H/a between the boost operator and the proper
+time Hamiltonian ˜H of the accelerated observer. For such an observer there is a thermal bath
+at (proper time) temperature T = 2π
+a .
+The other very well known example where symmetries again facilitate the derivation of the
+exact modular Hamiltonian is the case of conformal field theories (CFT) for spheres in any
+dimensions.
+For a CFT, Poincare symmetries are enlarged to the conformal group.
+These
+theories are characterized by having a traceless, symmetric and conserved stress tensor. This
+enlarges the number of conserved currents related to space-time symmetries which in general can
+be written as
+jµ = aν Tνµ + bαν xα Tνµ + c xν Tνµ + dα (x2gαν − 2 xαxν) Tνµ .
+(5)
+The corresponding conserved charges depend on parameters aµ, determining translations, the
+antisymmetric bµν, giving Lorentz transformations, c, related to dilatations, and dµ, for the so
+called special conformal transformations.
+Since there is a conformal transformation that maps the Rindler wedge to causal regions with
+spherical boundary, and the same transformation leaves the vacuum invariant for a CFT, then,
+the modular Hamiltonian is just the transformed Rindler modular Hamiltonian. It is easy to get
+K = 2π
+�
+|⃗x|<R
+dd−1x R2 − r2
+2R
+T00(⃗x) .
+(6)
+In this example, K is again local and proportional to T00, with a proportionality weight function
+β(r) ≡ R2−r2
+2R .
+Except for the two examples discussed above, the vacuum of a QFT in the Rindler wedge and
+the vacuum of a CFT in the sphere, there are only some other few known modular Hamiltonians,
+either local or not. The local ones in general are derived profiting from symmetry transformations
+that leave the state invariant. This is for example the case of the modular Hamiltonian for CFTs
+in 1 + 1 dimensions in presence of a global or local quench [20, 21, 22, 23, 24, 25, 26]. However,
+on general grounds, from the point of view of quantum information we do not expect locality to
+hold. In general, K will be given by a non local and non linear combination of the field operators
+at different positions inside the region.
+An example of a non local modular Hamiltonian which has been explicitly computed is the one
+for the vacuum state of the free massless fermion in d = 2 for several disjoint intervals [27, 28, 29].
+In this case K has a local term proportional to the energy density and an additional non local
+part given by a quadratic expression in the fermion field that connects in a very particular way
+points located in different intervals.
+In this paper we calculate the modular Hamiltonian for the vacuum state of non conformal
+(1 + 1) dimensional theories in the interval (0, R). These theories are defined in the semi infinite
+line, and result from the dimensional reduction of the d dimensional free massless scalar. Our
+strategy is to calculate the modular Hamiltonian of the reduced system by profiting of the known
+modular Hamiltonian of CFTs in spheres in any dimension.
+The free massless scalar in d space time dimensions can be dimensionally reduced to a sum
+of one dimensional theories, one for each angular mode.
+Since the reduction is obtained by
+integrating over the angular coordinates, these systems live in the semi infinite line. From the
+algebraic point of view, this is convenient when studying algebras assigned to spherical regions to
+calculate, for example, the entanglement entropy. In these coordinates, the local algebra assigned
+3
+
+R
+R
+Figure 1: The sphere of radius R corresponds to intervals of length R with one edge in the origin in
+the radial semi infinite line.
+to the region can be easily written in terms of fields φ(r, Ω) with nice localization properties. For
+example, points in the semi infinite line correspond to shells in the original space and intervals
+connected to the origin, to d-spheres (see figure 1). Concretely, in the radial coordinate, the
+canonical Hamiltonian for the massless free scalar decomposes as a sum over angular modes Hℓ⃗m
+H =
+�
+ℓ⃗m
+Hℓ⃗m .
+(7)
+with (ℓ⃗m) the angular mode label. In fact, there is a family of one dimensional Hamiltonian Hℓ⃗m
+for each dimension. In turn, the same decomposition occurs for the modular Hamiltonian (6)
+K =
+�
+ℓ⃗m
+Kℓ⃗m .
+(8)
+Taking into account that the vacuum state for a system composed by independent subsystems
+is a product of density matrices, here ρ = ⊗ρℓ⃗m, then it is immediate to identify the modular
+Hamiltonian mode Kℓ⃗m with the modular Hamiltonian of the one dimensional reduced system
+Hℓ⃗m. The Hamiltonian Hℓ⃗m does not correspond to a conformal relativistic theory due to an
+extra quadratic term proportional to 1/r2, whose proportionality constant depends on the di-
+mension of the original problem and the angular mode ℓ.
+Surprisingly, we find that Kℓ⃗m is
+still local and proportional to the energy density T001, with the same weight function β(r) that
+characterizes the modular Hamiltonian for CFTs in spheres. Our analytic results coincide with
+the suggested continuum limit of the entanglement Hamiltonian of blocks of consecutive sites in
+massless harmonic chains, recently studied in [30].
+This article is organized as follows.
+In section 2 we explicitly carry out the dimensional
+reduction. We write the scalar field in a basis of hyper-spherical harmonics, and after integrating
+out the angular coordinates we are left with a Hamiltonian for the reduced systems Hℓ⃗m of the
+form
+Hℓ⃗m = 1
+2
+�
+dr
+�
+�π2
+ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)
+r2
+�φ2
+ℓ⃗m
+�
+,
+(9)
+with
+µd(ℓ) = (d − 4)(d − 2)
+4
++ ℓ(ℓ + d − 3).
+(10)
+In section 3 the same procedure is followed to find the modular Hamiltonian
+Kℓ,⃗m = 2π
+�
+|⃗x|<R
+dr R2 − r2
+2R
+T ℓ,⃗m
+00 (⃗x) .
+(11)
+1Since translational invariance is lost, there is no conserved energy momentum tensor. The notation for the
+energy density is just a matter of convention.
+4
+
+In some way, the reduced theory, manifestly invariant under dilatations but non conformal, keeps
+the memory of the conformal symmetry of the parent d-dimensional theory [31, 32], with the
+same local modular Hamiltonian as that representing the vacuum of a CFT in a sphere. We delve
+into this in section 4, where we show that the reduced theories preserve an SL(2, R) symmetry,
+and that the modular transformation belongs to this subgroup. The modular Hamiltonian (11)
+written as a Noether charge can be correctly interpreted as the local operator implementing the
+modular flow.
+In section 5 we solve the spectrum of the modular Hamiltonian (11) and compute the entan-
+glement entropy in a segment connected to the origin. We find the analytic expression
+S(ℓ, d) = 1
+6 log R
+ϵ − iπ
+2
+� ∞
+0
+ds
+s
+sinh2(πs) log
+�4isΓ [is] Γ [−1 + d/2 + ℓ − is]
+Γ [−is] Γ [−1 + d/2 + ℓ + is]
+�
+,
+(12)
+which is logarithmically divergent, with coefficient 1/6 as expected for (1+1) theories, and has a
+constant term that depends both on the mode ℓ and the space time dimensions d of the original
+theory. Although the above integral cannot in general be solved analytically, we make some
+useful approximations to extract relevant information out of it. Moreover, by summing over ℓ we
+are able to recover the conformal anomaly in the logarithmic coefficient for the free scalar field
+in even dimensions, as well as the constant universal F term in d = 3. In doing the sum over
+the angular modes ℓ, we introduce a novel regularization implemented by a damping exponential
+exp[−ℓϵ/R], with the same cutoff ϵ that regularizes the radial coordinate r. This procedure
+generalizes the radial regularization scheme introduced by Srednicki in [33], where it is explicitly
+stated that for d ⩾ 4 regularization by a radial lattice turns out to be insufficient and the sum
+over partial waves does not converge. We end the discussion with some concluding remarks.
+2
+Spherical coordinates
+The free scalar action in spherical coordinates reads
+S = 1
+2
+�
+dtdrrd−2dΩ
+�
+−(∂0φ)2 + (∂rφ)2 − φ
+r2∆Sd−2φ
+�
+.
+(13)
+With the aim of reducing the above to a single integral in the radial direction, we Fourier
+transform the scalar field in the angular coordinates, using the real hyper-spherical harmonics
+as basis functions,
+φ(⃗r) =
+�
+ℓm1...md−3
+φℓm1...md−3(r)Y m1...md−3
+ℓ
+(ˆr),
+(14)
+with
+∆Sd−2Y m1...md−3
+ℓ
+(ˆr) = −ℓ(ℓ + d − 3)Y m1...md−3
+ℓ
+(ˆr),
+(15)
+�
+Sd−2 dΩY m1...md−3
+ℓ
+(ˆr)Y
+m′
+1...m′
+d−3
+ℓ′
+(ˆr) = δℓℓ′δm1m′
+1...δmd−3m′
+d−3.
+(16)
+After integrating the angular coordinates, we are left with
+S = 1
+2
+�
+ℓ⃗m
+�
+dtdrrd−2
+�
+−(∂0φℓ⃗m)2 + (∂rφℓ⃗m)2 + ℓ(ℓ + d − 3)
+r2
+φ2
+ℓ⃗m
+�
+.
+(17)
+5
+
+However, the theory looks simpler when defined in terms of the rescaled field �φℓ⃗m = r
+d−2
+2 φℓ⃗m,
+whose canonically conjugated momentum is �πℓ⃗m ≡ ∂0�φℓ⃗m,
+S = 1
+2
+�
+ℓ⃗m
+�
+dtdr
+�
+�−(∂0�φℓ⃗m)2 + rd−2
+�
+∂r
+� �φℓ⃗m
+r
+d−2
+2
+��2
++ ℓ(ℓ + d − 3)
+r2
+�φ2
+ℓ⃗m
+�
+� .
+(18)
+Functional variation with respect to the field leads to the equation of motion. Nevertheless, in
+order for the variational problem to be well posed we should impose specific boundary conditions
+at r = 0. In fact,
+δS =
+�
+ℓ⃗m
+��
+dtdr
+�
+∂2
+0 �φℓ⃗m −
+1
+r
+d−2
+2 ∂r
+�
+rd−2∂r
+� �φℓ⃗m
+r
+d−2
+2
+��
++ ℓ(ℓ + d − 3)
+r2
+�φℓ⃗m
+�
+δ�φℓ⃗m
++
+�
+dt
+�
+r
+d−2
+2 ∂r
+� �φℓ⃗m
+r
+d−2
+2
+�
+δ�φℓ⃗m
+������
+∞
+0
+�
+,
+(19)
+which requires either δ�φℓ⃗m(r = 0, t) = 0 (Dirichlet boundary conditions) or r
+d−2
+2 ∂r
+� �φℓ ⃗m
+r
+d−2
+2
+�
+→ 0
+(analogous to the ordinary Neumann boundary conditions). In the following we will adopt the
+former.
+The second term in (19) can be further simplified, which leads to the saddle point
+∂2
+0 �φℓ⃗m − ∂2
+r �φℓ⃗m + µd(ℓ)
+r2
+�φℓ⃗m = 0,
+(20)
+with
+µd(ℓ) = (d − 4)(d − 2)
+4
++ ℓ(ℓ + d − 3).
+(21)
+This partial differential equation can be solved by separation of variables, and expressed in terms
+of the original field φℓ⃗m, the radial eigenfunction problem is a Bessel equation, with solution
+jℓ(r) ≡
+1
+r(d−3)/2Jℓ+ d−3
+2 (kr). Therefore, the solution is
+�φℓ⃗m(t, r) = e±ikt√
+krJℓ+ d−3
+2 (kr),
+(22)
+which means that �φℓ⃗m ∼ rℓ+d/2−1 near kr ∼ 0, in agreement with the boundary conditions.
+Having stated that, it is also possible to rewrite the second term in (18) by getting rid of a
+boundary term2. More explicitly,
+S =
+�
+ℓ⃗m
+Sℓ⃗m
+(23)
+where
+Sℓ⃗m = 1
+2
+�
+dtdr
+�
+−(∂0�φℓ⃗m)2 + (∂r �φℓ⃗m)2 + µd(ℓ)
+r2
+�φ2
+ℓ⃗m
+�
+(24)
+can be thought of as the action for a free scalar living in the half line, satisfying Dirichlet
+boundary conditions at the origin. Note that, unlike the theory we started with, this is not a
+CFT because of the last term.
+2We would be able to ignore the boundary term provided �φ2
+ℓ⃗m went to zero faster than r. This is at least
+satisfied by the classical configuration (22).
+6
+
+The dimensional reduction of the free scalar Hamiltonian can be made following the same
+steps. But we can alternatively calculate the conserved charge due to time translations associated
+directly to the 1 + 1 dimensional action (24), yielding
+H = 1
+2
+�
+ℓ⃗m
+�
+dr
+�
+�π2
+ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)
+r2
+�φ2
+ℓ⃗m
+�
+(25)
+Once again we stress that �πℓ⃗m and �φℓ⃗m satisfy canonical commutation relations
+�
+�φℓ⃗m(r), �πℓ′ ⃗m′(r′)
+�
+= iδℓ,ℓ′δ⃗m,⃗m′δ(r − r′).
+(26)
+3
+The sphere modular Hamiltonian
+On the other hand, since the free scalar field theory in d spacetime dimensions is conformally
+invariant, when the whole system is in its ground state the modular Hamiltonian of a sphere is
+K = 1
+2
+�
+|x|<R
+dxd−1
+�R2 − r2
+2R
+�
+T00.
+(27)
+However, although the stress tensor involved in this expression must be traceless, the canonical
+stress tensor of the free scalar field is
+T (c)
+µν = ∂µφ∂νφ − 1
+2ηµν(∂φ)2,
+(28)
+which has non vanishing trace T µ
+µ = (1 − d/2) (∂φ)2 = (1−d/2)
+2
+∂2(φ2)3. Hence, it must be improved
+by adding a conserved symmetric tensor. A possible choice is
+T ′
+µν = T (c)
+µν − (1 − d/2)
+2(1 − d) (∂µ∂ν − ηµν∂2)φ2.
+(29)
+Therefore,
+K = 1
+2
+�
+|x|<R
+dxd−1
+�R2 − r2
+2R
+� �
+(∂0φ)2 + (∂iφ)2 − (1 − d/2)
+(1 − d) ∂2
+i φ2
+�
+.
+(30)
+Using the following identities:
+(∂iφ)2 = (∂rφ)2 − φ
+r2∆Sd−2φ,
+(31)
+where we have partially integrated the angular piece, and
+∂2
+i φ2 =
+1
+rd−2∂r
+�
+rd−2∂rφ2�
++ 1
+r2∆Sd−2φ2,
+(32)
+we arrive at
+K = 1
+2
+�
+ℓ⃗m
+�
+drrd−2
+�R2 − r2
+2R
+� �
+π2
+ℓ⃗m + (∂rφℓ⃗m)2 + ℓ(ℓ + d − 3)
+r2
+φ2
+ℓ⃗m−
+−(1 − d/2)
+(1 − d)
+�
+∂2
+rφ2
+ℓ⃗m + d − 2
+r
+∂rφ2
+ℓ⃗m
+��
+,
+(33)
+3This identity holds on-shell.
+7
+
+In terms of the canonically conjugated operators,
+K = 1
+2
+�
+ℓ⃗m
+�
+dr
+�R2 − r2
+2R
+� �
+�π2
+ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)
+r2
+�φ2
+ℓ⃗m−
+− d − 2
+2(d − 1)
+�
+3∂r
+� �φ2
+ℓ⃗m
+r
+�
++ r∂2
+r
+� �φ2
+ℓ⃗m
+r
+���
+,
+(34)
+Note that the second line of (34), together with the prefactor (R2 − r2), is a total derivative in
+disguise. Hence,
+K = 1
+2
+�
+ℓ⃗m
+�� R
+0
+dr
+�R2 − r2
+2R
+� �
+�π2
+ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)
+r2
+�φ2
+ℓ⃗m
+�
+−
+− d − 2
+2(d − 1)
+�
+(R2 − r2)
+2R
+r∂r
+� �φ2
+ℓ⃗m
+r
+�
++ R
+� �φ2
+ℓ⃗m
+r
+������
+R
+0
+�
+,
+(35)
+The boundary terms (coming from the improving) can be interpreted in general, as an ambiguity
+in the definition of modular Hamiltonian in a region, and safely ignored as explained in [34].
+Consequently, the modular Hamiltonian of the d dimensional free scalar is
+K =
+�
+ℓ⃗m
+Kℓ⃗m,
+(36)
+where
+Kℓ⃗m = 1
+2
+� R
+0
+dr
+�R2 − r2
+2R
+� �
+�π2
+ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)
+r2
+�φ2
+ℓ⃗m
+�
+(37)
+can be interpreted as the modular Hamiltonian for the vacuum of (24) in a segment.
+This
+identification rests on the fact that the theory decomposes into independent sectors, labeled by
+the angular modes, so the state must write as the direct product of the states pertaining to each
+sector. But, most remarkably, this modular Hamiltonian is still local in the energy density. In
+other words, (37) agrees with the general expression (27) in spite of the reduced one dimensional
+theories being non conformal. In the next section we analyse this in detail, paying attention to
+the symmetries which survive the dimensional reduction.
+Provided that (37) defines the reduced state of a free field theory, Wick’s theorem guarantees
+it can be expressed in terms of the two-point correlators. In fact, for a Gaussian state with
+modular Hamiltonian
+K =
+�
+V
+dd−1x1dd−1x2 [φ(x1)M(x1, x2)φ(x2) + π(x1)N(x1, x2)π(x2)] ,
+(38)
+and correlators
+X = ⟨φ(x1)φ(x2)⟩ ,
+P = ⟨π(x1)π(x2)⟩ ,
+(39)
+the following relation must be satisfied [35]4
+M.X = P.N
+(40)
+4Here the product is a bi-local function constructed as
+[M.X] (x1, x2) ≡
+�
+V
+dyM(x1, y)X(y, x2)
+8
+
+In the case at hand,
+M(r, r′) = −2πδ(r − r′)
+�
+β(r)∂2
+r + ∂rβ(r)∂r − β(r) µ
+r2
+�
+(41)
+and
+N(r, r′) = 2πδ(r − r′)β(r) .
+(42)
+Meanwhile, the explicit form of the correlators for the one dimensional theory (24) is [36]
+X(r1, r2) =
+Γ [ℓ + d/2 − 1]
+2Γ
+� 1
+2
+�
+Γ
+�
+ℓ + d−1
+2
+�
+�r1
+r2
+�ℓ+ d
+2 −1
+2F1
+�
+1
+2, ℓ + d
+2 − 1; ℓ + d − 1
+2
+;
+�r1
+r2
+�2�
+,
+(43)
+P(r1, r2) =
+2Γ(ℓ + d/2)
+Γ
+� 1
+2
+�
+Γ
+�
+ℓ + d−1
+2
+�
+(r2
+2 − r2
+1)
+�r1
+r2
+�ℓ+ d
+2 −1 �
+A 2F1
+�
+1
+2, ℓ + d
+2; ℓ + d − 1
+2
+;
+�r1
+r2
+�2�
++B 2F1
+�
+−1
+2, ℓ + d
+2; ℓ + d − 1
+2
+;
+�r1
+r2
+�2��
+,
+(44)
+where A =
+�
+ℓ + d−1
+2
+�
+(1 − r2
+1/r2
+2) − 1, B = 1 − ℓ − d/2, and r1 < r2.
+Using these concrete
+expressions it is possible to check that (40) indeed holds.
+4
+Symmetries
+The locality of (37) suggests the existence of a symmetry with a conserved current such that the
+modular Hamiltonian is the corresponding Noether charge. This has to be an endomorphism in
+the causal wedge of the region, and must point in the time direction at t = 0. For CFTs in spheres
+in any dimensions, this is the conformal transformation that maps the spherical boundary in
+itself. For an interval (0, R) in the half line, whose causal wedge is a half diamond, the symmetry
+transformation leaves the boundary point r = R fixed. The identification of this symmetry in
+the present case is the natural path to justify the locality of (37). With this aim, we first discuss
+the symmetries of the reduced theories with action (24).
+The symmetries of (24) are a subgroup of the conformal transformations inherited from higher
+dimensions, in particular those which involve only the time and radial coordinates, and that map
+the line r = 0 into itself. These are
+• Time translations:
+t → t + t0
+(45)
+• Dilatations:
+(t, r) → (λt, λr)
+(46)
+• Special conformal transformations with parameter bµ = α
+Rˆeµ
+t :
+(t, r) →
+�
+tR2 + αR(t2 − r2)
+R2 + 2αRt + α2(t2 − r2),
+rR2
+R2 + 2αRt + α2(t2 − r2)
+�
+.
+(47)
+Infinitesimally, that is, if we set α = ϵ << 1, then
+(t, r) →
+�
+t − ϵ(t2 + r2)/R, r − 2ϵtr/R
+�
+.
+(48)
+9
+
+The generators of the transformations listed above are P0 = i∂t, D = i(t∂t + r∂r), and K0 =
+i ((t2 + r2)∂t + 2tr∂r) respectively. These close an sl(2, R) algebra, which can be expressed in a
+more suggestive way identifying L−1 ≡ P0, L0 ≡ D, L1 ≡ K0, so that
+i [Lm, Ln]LB = (m − n)Lm+n.
+(49)
+Just for completeness, we note that one would have expected the original conformal group
+SO(d, 2) to break into SO(2, 2) ∼ SL(2, R) ⊗ SL(2, R) [31, 32], with six generators. In fact,
+besides the three generators already mentioned, there are three more that do not mix the angular
+coordinates with (t, r), associated to
+• Translations in the radial direction:
+r → r + r0
+(50)
+• Boosts:
+(t, r) → (t + ϵr, r + ϵt)
+(51)
+• Special conformal transformations with parameter bµ = α
+Rˆeµ
+r
+(t, r) →
+�
+tR2
+R2 − 2αRr − α2(t2 − r2),
+rR2 + αR(t2 − r2)
+R2 − 2αRr − α2(t2 − r2)
+�
+.
+(52)
+These are ˆeµ
+rPµ, ˆeµ
+rM0µ and ˆeµ
+rKµ, respectively. However, it is easy to see that they fail to become
+symmetries of the dimensionally reduced theory.
+Then, the modular symmetry of the reduced theories we are looking for must be a particular
+composition of the identified symmetry transformations (45) - (47).
+On the other hand, we know that the modular symmetry for the parent conformal theory is
+associated to the generator of the boosts as seen from the domain of dependence of the ball [37],
+ζ = π
+R
+�
+(R2 − t2 − |⃗x|2)∂t − 2txi∂i
+�
+.
+(53)
+In fact, comparing with (45) and (47), we notice that this transformation in the semi infinite line
+is the composition of a time translation of parameter ϵπR and a special conformal transformation
+of parameter ϵ π
+R. Let us check this explicitly.
+In spherical coordinates, the infinitesimal transformation reads
+t −→ t′ = t + ϵ π
+R(R2 − t2 − r2)
+r −→ r′ = r + ϵ π
+R(−2tr)
+Ω −→ Ω′ = Ω
+(54)
+Since the invariance of the kinetic term is guaranteed, we need only to check the invariance of
+the quadratic term dtdr/r2, which is less evident. On the one hand, we have
+dtdr
+=
+dt′dr′
+����
+∂t
+∂t′
+∂t
+∂r′
+∂r
+∂t′
+∂r
+∂r′
+���� = dt′dr′
+����
+1 + 2πϵt′/R + O(ϵ2)
+2πϵr′/R + O(ϵ2)
+2πϵr′/R + O(ϵ2)
+1 + 2πϵt′/R + O(ϵ2)
+����
+∼
+dt′dr′(1 + 4πϵt′/R).
+(55)
+10
+
+On the other hand,
+1
+r2 ∼
+1
+(r′ + 2πϵt′r′/R)2 = 1
+r′2(1 − 4πϵt′/R + O(ϵ2)).
+(56)
+Hence,
+dtdr
+r2
+= dt′dr′
+r′2
++ O(ϵ2).
+(57)
+By Noether’s theorem, there must exist a conserved current associated to (54), which is of the
+form5
+jµ =
+�
+δL
+δ(∂µφ)∂νφ − Lδµ
+ν
+�
+ζν,
+(58)
+or, in components,
+jt = 1
+2
+�
+(∂tφ)2 + (∂rφ)2 + µ
+r2φ2� (R2 − t2 − r2)
+R
+− 2tr
+R∂rφ∂tφ,
+(59)
+jr = 1
+2
+�
+(∂tφ)2 + (∂rφ)2 − µ
+r2φ2�
+2tr
+R − ∂rφ∂tφ(R2 − t2 − r2)
+R
+.
+(60)
+Finally, the current above corresponds to a modular Hamiltonian
+Kℓ⃗m =
+� R
+0
+drj0(t = 0, r)
+= 1
+2
+� R
+0
+dr
+�R2 − r2
+2R
+� �
+�π2
+ℓ⃗m + (∂r �φℓ⃗m)2 + µd(ℓ)
+r2
+�φ2
+ℓ⃗m
+�
+,
+(61)
+the same as (37) deduced in the previous section from different arguments.
+5
+Modular Hamiltonian and entropy
+In this section we study the spectrum of the modular Hamiltonian (37). Solving the eigenfunction
+problem allows us to compute the entanglement entropy for an interval attached to the origin,
+as a function of the angular mode ℓ and the original spacetime dimension d. Then we sum over
+the modes and compare the result with the entanglement entropy of the d-sphere.
+5.1
+Eigenfunctions
+In general, given a quadratic modular Hamiltonian of a region V , of the form
+K =
+�
+V
+dd−1x dd−1x′ (φ(x)M(x, x′)φ(x′) + π(x)N(x, x′)π(x′)) ,
+(62)
+with M and N real symmetric operators, the eigenfunctions are those of the right and left action
+of M.N, namely
+(N.M)us = s2us
+(63)
+(M.N)vs = s2vs.
+(64)
+5We remove the tildes and the angular mode labels to avoid cluttering.
+11
+
+This leads to the alternative way of writing K
+K =
+�
+V
+dd−1x
+� ∞
+0
+ds us(x) s v∗
+s(x).
+(65)
+More concretely, the problem we are interested in is defined by (41) and (42), so the eigenfunctions
+u and v satisfy the following hypergeometric equations6
+�
+β2∂2
+r + β∂rβ∂r − β2 µ
+r2
+�
+us = −s2us
+(66)
+�
+β2∂2
+r + 3β∂rβ∂r +
+�
+β∂2
+rβ + (∂rβ)2 − β2 µ
+r2
+��
+vs = −s2vs
+(67)
+The solutions of these equations are7
+us(r) = Nu
+� r
+R
+�−1+ d
+2 +ℓ �R2 − r2
+R2
+�−is
+2F1
+�1
+2 − is, −1 + d
+2 + ℓ − is, d
+2 − 1
+2 + ℓ, r2
+R2
+�
+vs(r) = Nu
+R
+β(r)us(r),
+(68)
+where Nu is a normalization constant.
+Near r ∼ 0 the solutions behave as,
+us(r) ∼ vs(r) ∝ r−1+ d
+2 +ℓ
+(69)
+in agreement with the classical profile (22), whereas near r ∼ R they behave as
+us(r) ∼ Nu
+��R − r
+R
+�−is
+α(s) + c.c.
+�
+vs(r) ∼ Nu
+��R − r
+R
+�−1−is
+α(s) + c.c.
+�
+,
+(70)
+with
+α(s) =
+2−isΓ
+� d−1
+2 + ℓ
+�
+Γ [2is]
+Γ
+�
+is + 1
+2
+�
+Γ
+�
+is + ℓ + d
+2 − 1
+�.
+(71)
+It is very important to keep in mind that there is a branch point at r = R. In fact, since the
+eigenfunctions must satisfy the orthogonality relation
+� R
+0
+drus(r)v∗
+s′(r) = δ(s − s′) ,
+(72)
+in order to find out the normalization factor Nu we substitute in (72) the leading terms in their
+Taylor series expansion (70), because only the region near r ∼ R can contribute with a Dirac
+delta function. That results in
+� R
+0
+drus(r)v∗
+s′(r) ∼ 2|Nu|2Re [I(s − s′)α(s)α∗(s′) + I(s + s′)α(s)α(s′)] ,
+(73)
+6For later convenience we renormalize the eigenvalues to absorb a factor 1/(2π)2.
+7There is an additional independent solution, but we dismiss it because it does not go to zero at r = 0, as
+mandated by the boundary conditions.
+12
+
+where
+I(s) ≡
+� R
+0
+dr
+R
+R − r exp
+�
+−is log
+�R − r
+R
+��
+= R
+� i
+s + πδ(s)
+�
+.
+(74)
+Hence, neglecting the finite terms8, we have that (72) holds provided that
+Nu =
+1
+√
+2πR|α(s)|
+,
+(75)
+save an overall phase that we set to one for convenience.
+5.2
+The entropy
+As explained in [38] in the context of the free chiral scalar, we can take advantage of the orthogo-
+nality relation to simplify the computation of the entanglement entropy, which can be expressed
+as a regularized integral over a small region behind the end point r = R, of the form
+S(ℓ, d) =
+� R−ϵ
+0
+dr
+� ∞
+0
+ds us(r)g(s)v∗
+s(r)
+= − lim
+δs→0
+� R
+R−ϵ
+dr
+� ∞
+0
+ds us(r)g(s)v∗
+s+δs(r) ,
+(76)
+with
+g(s) = 1 + coth(πs)
+2
+log
+�1 + coth(πs)
+2
+�
++ 1 − coth(πs)
+2
+log
+�1 − coth(πs)
+2
+�
+(77)
+Note that since we expect the entanglement entropy of a QFT to diverge due to the short range
+correlations between modes at both sides of the boundary, we regularized it by introducing a
+small UV cutoff ϵ. Furthermore, in going from the first to the second line of (76) we shifted the
+v sub index, summing over slightly off diagonal elements. For fixed δs ̸= 0 the integral defined
+on the whole interval vanishes because of (72), leading to an integral just behind the boundary.
+This trick allows us to substitute the expansion (70), which is much easier to integrate than the
+original solutions (68). Finally, we get
+S(ℓ, d) = 1
+6 log R
+ϵ − 1
+π
+� ∞
+0
+ds g′(s)Arg(α(s)),
+(78)
+or, more explicitly,
+S(ℓ, d) = 1
+6 log R
+ϵ − iπ
+2
+� ∞
+0
+ds
+s
+sinh2(πs) log
+�4isΓ [is] Γ [−1 + d/2 + ℓ − is]
+Γ [−is] Γ [−1 + d/2 + ℓ + is]
+�
+(79)
+The logarithmic coefficient 1/6 is the expected result for a (1+1) dimensional theory. Meanwhile,
+the constant term is expressed in terms of an integral that cannot be solved explicitly. For later
+8We also neglect a contribution of the form δ(s + s′), coming from the second term in (73), because it is
+non-zero only at s = s′ = 0.
+13
+
+0
+20
+40
+60
+80
+100
+-0.8
+-0.6
+-0.4
+-0.2
+0.0
+Figure 2: Constant term of the entropy at d = 3, as a function of the angular mode ℓ. The red dots represent
+the exact numerical value of (81), for ℓ = {1, 2, 5, 10, 15, 20, 30, 40, 100}. The blue curve corresponds to the fit
+f(ℓ, d = 3) = c0 + c1 log ℓ, with c0 = 1.345 × 10−5 and c1 = −0.1666.
+convenience, we write it as a sum of two contributions, one that does not depend neither on the
+dimension nor on the angular mode
+c ≡ −iπ
+2
+� ∞
+0
+ds
+s
+sinh2(πs) log
+�4isΓ [is]
+Γ [−is]
+�
+,
+(80)
+and another which does depend on both parameters
+f(ℓ, d) ≡ −iπ
+2
+� ∞
+0
+ds
+s
+sinh2(πs) log
+�Γ [−1 + d/2 + ℓ − is]
+Γ [−1 + d/2 + ℓ + is]
+�
+(81)
+Although it is unfortunately impossible to find an analytic expression for the integral, for
+sufficiently large modes we can make use of the Stirling’s approximation
+log Γ(z) ∼ z log z − z + 1
+2 log 2π
+z +
+N−1
+�
+n=1
+B2n
+2n(2n − 1)z2n−1,
+|z| → ∞
+(82)
+to write
+log
+�Γ [−1 + d/2 + ℓ − is]
+Γ [−1 + d/2 + ℓ + is]
+�
+∼ −2is log ℓ +
+∞
+�
+k=2
+⌊ k+1
+2 ⌋
+�
+m=1
+(k − 2)!
+ℓk−1
+ak,m(s)
++ 1
+2
+∞
+�
+k=1
+⌊ k+1
+2 ⌋
+�
+m=1
+(k − 1)!
+ℓk
+ak,m(s) +
+∞
+�
+n=1
+∞
+�
+k=1
+⌊ k+1
+2 ⌋
+�
+m=1
+B2n(2n + k − 2)!
+(2n)!ℓ2n+k−1
+ak,m(s),
+ℓ >> 1,
+(83)
+where
+ak,m(s) =
+2i(−1)k+m
+(2m − 1)!(k + 1 − 2m)!
+�
+−1 + d
+2
+�k+1−2m
+s2m−1.
+(84)
+This means that the constant term grows logarithmically with the mode ℓ, with corrections that
+decay as positive powers of 1/ℓ. In fact, performing the integration over the variable s order by
+order in the expansion, we can straightforwardly check that the first few leading terms read
+f(ℓ, d) ∼ −1
+6 log ℓ + a1
+ℓ + a2
+ℓ2 + a3
+ℓ3 + a4
+ℓ4 + O
+� 1
+ℓ5
+�
+,
+(85)
+14
+
+0
+5
+10
+15
+20
+25
+-0.6
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+Figure 3: ∆f(ℓ, 3) ≡ f(ℓ, 3) − f(ℓ = 1, 3). Blue: direct numerical integration of (81). Orange: calculation with a
+radial lattice regularization. ℓ = {1, 2, 5, 10, 20}
+with
+a1
+=
+1
+4 − d
+12
+(86)
+a2
+=
+7
+40 − d
+8 + d2
+48
+(87)
+a3
+=
+3
+20 − 7d
+40 + d2
+16 − d3
+144
+(88)
+a4
+=
+73
+560 − 9d
+40 + 21d2
+160 − d3
+32 + d4
+384
+(89)
+Quite surprisingly, the logarithmic term already approximates f(ℓ, d) at ℓ ∼ O(1) very accurately,
+as shown in figure (2).
+In figure (3) we compare the numerical value of (81) with the one obtained from direct calcu-
+lation in a radial lattice, again at d = 3. Since the constant term depends on the regularization
+scheme, we subtract the one corresponding to ℓ = 1 and compare ∆f(ℓ, 3) ≡ f(ℓ, 3)−f(ℓ = 1, 3).
+Although it is very hard to achieve good precision in the lattice9, we find reasonable agreement.
+For example, for ℓ = 10, numerical integration yields ∆f = −0.3737, while the lattice computa-
+tion gives ∆f = −0.3795.
+5.3
+Recovering the scalar entropy
+As discussed in section 3, the modular Hamiltonian of the free scalar in the sphere is equal to
+the sum over ℓ of the modular Hamiltonian pertaining to each one dimensional theory in the
+segment. Consequently, we expect that summing (79) must necessarily reproduce the general
+structure for the entanglement entropy,
+S =
+�
+#
+� R
+ϵ
+�d−2 + ... + clog log R
+ϵ ,
+d
+even
+#
+� R
+ϵ
+�d−2 + ... + F,
+d
+odd
+(90)
+that is, an infinite contribution controlled by the area term, and a universal piece, either in
+the form of a logarithmic coefficient in even dimensions, which is precisely the trace anomaly
+9Roughly speaking, the value of ℓ gives a lower bound for the meaningful radios R/ϵ >> ℓ
+15
+
+coefficient associated to the Euler density [39, 40, 41], or a constant term in odd dimensions
+[42, 43, 44].
+To show that this indeed holds, we need to introduce a cutoff in ℓ to regularize the sum. More
+concretely, we introduce a damping exponential so that
+S =
+∞
+�
+ℓ=0
+λ(ℓ, d)S(ℓ, d)e−ℓϵ/R.
+(91)
+where
+λ(ℓ, d) = (2ℓ + d − 3)(ℓ + d − 4)!
+ℓ!(d − 3)!
+(92)
+is the density of states. Note that this grows as ℓd−3 for ℓ >> 1.
+Given the complicated expression of the constant term f(ℓ, d), we approximate it by its large
+ℓ expansion, leading to
+S =
+∞
+�
+ℓ=1
+λ(ℓ, d)
+�
+1
+6 log R
+ϵ + c − 1
+6 log ℓ +
+jmax
+�
+j=1
+aj
+ℓj
+�
+e−ℓϵ/R + λ(0, d)S(0, d) + correction.
+(93)
+The correction above accounts for the error made when approximating f(ℓ, d) by its series ex-
+pansion, truncated at O(ℓ−jmax).
+It is straightforward to verify that (93) reproduces (90). For example, the divergent pieces
+come from terms with the general structure
+∞
+�
+ℓ=1
+ℓp
+�
+log R
+ϵ − log ℓ
+�
+e−ℓϵ/R = −Γ′(p + 1)
+�R
+ϵ
+�p+1
++ ζ(−p) log R
+ϵ + ζ′(−p),
+(94)
+∞
+�
+ℓ=1
+ℓpe−ℓϵ/R = p!
+�R
+ϵ
+�p+1
++ ζ(−p),
+(95)
+with {p | p ∈ N0 ∧ p ≤ d − 3}, and
+∞
+�
+ℓ=1
+1
+ℓe−ℓϵ/R = log R
+ϵ .
+(96)
+Note that the logarithmic term, only present in even dimensions, stems from (94) and (96).
+Based on this observation, it is worth pointing out that in order to compute the logarithmic
+coefficient we only need to take into account the first d − 1 terms in the expansion of f(ℓ, d),
+that is, jmax = d − 2. Subleading corrections give finite contributions at most.
+Just to explicitly address some relevant specific cases, at d = 6 we get
+clog(d = 6) = 29
+540 + a1 + 13
+6 a2 + 3
+2a3 + 1
+3a4,
+(97)
+and, substituting (86), (87), (88), (89),
+clog(d = 6) =
+1
+756,
+(98)
+16
+
+10
+20
+30
+40
+50
+60
+70
+5.19
+5.18
+5.185
+5.195
+corr
+Figure 4: Error made in the computation of the constant term F when approximating f(ℓ, 3) by − 1
+6 log ℓ + a2
+ℓ2 .
+ℓmax is the greatest angular momentum that is summed over. We see that the correction converges very fast to
+∼ 0.00519641
+in agreement with the expected anomaly value. On the other hand, at d = 4 we get
+clog(d = 4) = 1
+18 + a1 + 2a2,
+(99)
+which leads to the expected anomaly coefficient
+clog(d = 4) = − 1
+90.
+(100)
+The case of d = 3 is different from the ones discussed above in that it has no logarithmic term
+and the universal piece in the entanglement entropy is associated to the constant term F. In
+fact, direct calculation yields, using (86)
+clog(d = 3) = 2a1 = 0.
+(101)
+Regarding the constant F, the infinite tail in the 1/ℓ expansion must in principle be taken into
+account. For that reason, we regularize the sum taking up to jmax = 2 and then add a finite
+contribution which corrects the approximation, giving the exact value for the constant term in
+the series of f(ℓ, d). That is,
+correction = 2
+lim
+ℓmax→∞
+ℓmax
+�
+ℓ=1
+�
+f(ℓ, d = 3) + 1
+6 log ℓ − a2
+ℓ2
+�
+(102)
+In figure (4) we plot the above correction as a function of ℓmax and show that it converges very
+fast to
+correction ∼ 0.00519641
+(103)
+According to (93), another term which contributes is
+f(0, 3) = −iπ
+2
+� ∞
+0
+ds
+s
+sinh2(πs) log
+�Γ [1/2 − is]
+Γ [1/2 + is]
+�
+∼ 0.278435.
+(104)
+Gathering all the pieces together, we finally get
+F = π2
+3 a2 + ζ′(0)
+3
++ f(0, 3) + correction ∼ −0.0638049,
+(105)
+which is within 0.003% of the exact value [42, 44]. Note that the constant c does not contribute
+at all to F.
+17
+
+6
+Final remarks
+In this article, we focused on theories in the semi infinite line constructed from the dimensional
+reduction of a free scalar in d dimensions. Given that the decomposition of the parent theory H
+into independent sectors Hℓ⃗m, labeled by the angular modes, also holds for the vacuum modular
+Hamiltonian in spheres K = �
+ℓ⃗m Kℓ⃗m, and provided that the vacuum state of the system is
+the product ρ = ⊗ρℓ⃗m, then, it is immediate to identify the modular Hamiltonian mode Kℓ⃗m
+with the modular Hamiltonian of the one-dimensional reduced system Hℓ⃗m in the interval (0, R).
+Remarkably, the resulting modular Hamiltonian is local and proportional to the energy density,
+with the same weight function β(r) = R2−r2
+2R
+as the one characteristic of a CFT in a sphere R.
+We complemented the previous analysis with the study of the symmetries inherited from the
+d dimensional conformal theory. This approach evidences the fact that the symmetry behind
+the locality of the reduced modular Hamiltonian is just the restriction to the semi infinite line of
+the original modular symmetry in d dimensions. We identified the conserved current associated
+to this symmetry transformation and checked that the Kℓ⃗m found by dimensional reduction
+coincides with the Noether charge.
+On the other hand, the spectral decomposition of the modular Hamiltonian leads to an analytic
+expression for the corresponding entanglement entropy (EE) which in turn, after summing over
+the angular modes, allowed us to recover the EE of the original d dimensional theory in the sphere.
+To make sense of the sum, we used a novel regularization implemented by a damping exponential
+parametrized by the same cutoff ϵ that regularizes the radial coordinate. As we mentioned in
+the introduction, in a way, this procedure generalizes the one introduced by Srednicki in [33]
+and provides an additional tool to calculate analytically the EE logarithmic coefficient in even
+dimensions, the universal constant term in d = 3, among others.
+It would certainly be interesting to explore in the future the modular Hamiltonian of non-
+conformal theories constructed from the dimensional reduction of other free theories, fermions
+for example. We expect that the decomposition of the parent theory into independent sectors
+must carry over unaltered, as well as the symmetry arguments which justify the resulting modular
+Hamiltonian is a conserved charge.
+Acknowledgments
+We thank H.Casini, C.Fosco, E.Tonni and G.Torroba for discussions while this work was being
+carried out. This work was supported by CONICET, CNEA and Universidad Nacional de Cuyo,
+Instituto Balseiro, Argentina.
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+

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+CONSIP: Consistency Protocol for Hopping
+Function Exchange and Black listing in TSCH
+Stefano Scanzio, Federico Bitondo, Gianluca Cena, and Adriano Valenzano
+National Research Council of Italy (CNR–IEIIT), Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
+Email: [email protected], [email protected],
+[email protected], [email protected]
+Abstract—The use of white and black listing techniques in
+Wireless Sensor Networks (WSN), and in particular those which
+are based on the Time Slotted Channel Hopping (TSCH) op-
+erating mode of IEEE 802.15.4, permits to improve reliability
+and latency by performing transmissions on the best channels.
+Techniques that operate on a per-link basis are deemed quite
+effective, but proper operation requires that the two end points
+involved in the communication agree on the channels to be used
+for transmission. On the contrary, communication in the network
+can be prevented, eventually leading, in the worst cases, to the
+disconnection of part of the nodes.
+This paper presents CONSIP, a technique aimed to ensure
+strict consistency in the information exchanged between the
+nodes and used to drive communication, by preventing a priori
+the aforementioned problem from occurring. Results show a
+slight increase in energy consumption, due to the use of a
+backup cell, whereas communication latency does not worsen.
+The effectiveness of CONSIP was assessed by means of an
+experimental campaign, and the only drawback we found is that
+the backup cell, which is required to be reserved per link, may
+limit the number of nodes in dense networks.
+I. INTRODUCTION
+The recent evolution of factories can be viewed as a sort
+of new, pacific futurism movement1, in which characteristics
+such as speed, simultaneity, dynamism, and innovation play a
+relevant role in next-generation industrial networks and plants.
+On the one hand, the Industry 4.0 [1], [2] revolution (and
+beyond) is an example of this transition, where communication
+networks are becoming more and more heterogeneous in terms
+of the employed technologies [3] and wireless extensions are
+playing an increasingly important role for enabling the afore-
+mentioned characteristics. On the other hand, the (Industrial)
+Internet of Things (IoT/IIoT) paradigm [4], [5], [6] enables
+autonomous communication between network devices, and is
+boosting the need of wireless technologies with more specific
+features, able to meet increasingly demanding requirements.
+In this direction, the set of technologies collectively known
+with the term Wireless Sensor (and Actuator) Networks
+(WSN/WSAN) is extensively used to collect data and, some-
+times, to perform actuations in distributed systems, where
+devices are typically battery-powered and the amount of data
+to be exchanged is quite small. Among the available solutions,
+the Deterministic and Synchronous Multichannel Extension
+(DSME) and the Time Slotted Channel Hopping (TSCH)
+1Futurism was a social and artistic movement, which originated in Italy in
+the early 20th century.
+operating modes of the IEEE 802.15.4 standard [7] show
+interesting features and are becoming quite popular. In both
+modes, the transmissions of packets in a data stream, as well
+as the retransmissions of the same packet, are carried out auto-
+matically at the MAC layer on different frequencies/channels,
+which consequently makes the quality of the communication
+link, as seen by network nodes, more stable.
+This work focuses on TSCH, and in particular it is related
+to those techniques, known as black listing and white listing,
+that are aimed at increasing the quality of communication, by
+removing the worst channels or by selecting the best channels,
+respectively. For example, if a network node is given the ability
+to select the best channels to perform its transmissions, it will
+consequently experience improvements in terms of reliability,
+latency, and power consumption. To this end, many black
+and white listing algorithms were proposed in the scientific
+literature, and their typical operations can be subdivided into
+three basic steps:
+1) evaluation: inferring how each channel will likely be-
+have in the near future, typically using statistics col-
+lected from the recent past;
+2) selection: deciding whether, where, and how to use these
+channels, that is, planning a selection strategy;
+3) propagation: delivering the channel selection strategy to
+the involved nodes in a consistent way.
+The main content of this paper is related to the last point.
+In particular, it is about ensuring that, at any given time, the
+sender and receiver nodes on a link use the same channel for
+communicating. In fact, if this property was not guaranteed,
+communication within the network could be prevented, possi-
+bly with severe consequences, which typically consist in the
+disconnection, sometimes permanent, of one or more nodes. To
+this extent we proposed the CONSIP protocol, which ensures
+to the involved nodes a reliable exchange of the information
+about the channels to be used in future transmissions. It
+grounds on the idea to leave two disjoint communications links
+open during this exchange, which virtually operate in parallel,
+the former based on the old information about channels and
+the second using the new one. Only when both nodes are
+certain that they have agreed to use the new information, the
+previous communication based on the old channel selection
+is deactivated. An extensive experimental campaign based
+on simulation has been carried out, in order to evaluate the
+This is the author’s version of an article that has been published in this journal.
+Changes were made to this version by the publisher prior to publication.
+The final version of record is available at https://doi.org/10.1109/WFCS53837.2022.9779192
+Copyright (c) 2022 IEEE. Personal use is permitted.
+For any other purposes, permission must be obtained from the IEEE by emailing [email protected].
+arXiv:2301.00070v1  [cs.NI]  30 Dec 2022
+
+CONSIP functionality from the point of view of a number
+of statistical indicators, and in particular power consumption,
+which is a very important performance metric for this kind of
+networks.
+In the following, the concepts behind white and black listing
+are analyzed and described together in Section II, with an
+extensive analysis of the state of the art. The CONSIP protocol
+is firstly presented intuitively, and then formally, in Section III.
+Section IV describes the simulator and the parameters we used
+in the simulation, including the energy model, while results are
+reported in Section V, which precedes Section VI that draws
+some concluding remarks.
+II. BLACK AND WHITE LISTING
+The channel hopping technique enables nodes in a wireless
+network to periodically change the transmission frequency
+of links in order to mitigate the effects of disturbance and
+interference on the quality of communication. The ideas be-
+hind channel hopping were proposed more than one decade
+ago [8], and currently they are adopted in several network
+technologies like WirelessHART [9], Bluetooth [10], and
+TSCH. In particular, time in TSCH is divided in timeslots
+of equal length [11], [12], while channel hopping [8] selects
+the effective transmission channel through a pseudo-random
+function ν shared between the transmitter and the receiver.
+In this protocol, traffic is scheduled by reserving timeslots, to
+permit nodes to be switched off when no data exchange is
+scheduled for them, consequently saving energy.
+A scheduled transmission is identified by a pair of values,
+the slot offset (o) and the channel offset (c), which iden-
+tify a position in a matrix of dimension Nslots × Nch. The
+schedule repeats over time with period Nslots slots. Typically
+Nslots = 101, and the slot duration is equal to 20 ms. As a
+consequence, the repetition period of the slotframe (i.e., of
+the matrix) is 2.02 s. Each slot is identified by a (practically)
+unique and strictly increasing unsigned integer number x,
+which is known as the Absolute Slot Number (ASN). In every
+slot up to Nch transmissions can be performed at the same
+time on distinct links, each one related to a different channel
+offset c ∈ {1, ..., Nch}.
+The shared hopping function ν returns the physical channel
+ch used for the actual transmission, and can be expressed as
+ch = ν(x, c) ≜ H((x + c) mod Nch),
+(1)
+where H(i) is the so called hopping sequence. Given an
+integer value i ∈ {0, ..., Nch − 1}, H(i) basically returns
+the element in position i of an array of dimension Nch,
+which encodes the physical channel. If protocol parameters
+are set in such a way that two subsequent transmissions of
+the same data frame are not spaced by a multiple of Nch (as
+typically happens in real networks), then they will take place
+on different physical channels. The hopping sequence H(i)
+is usually chosen so that subsequent transmissions take place
+on channels that are spaced wide enough. In this way, retries
+of the same packet are unlikely to suffer repeatedly from the
+effect of the same source of interference [13]. For instance,
+in the 2.4 GHz band a single Wi-Fi channel can span over
+multiple IEEE 802.15.4 channels that use the O-QPSK PHY.
+Channel hopping has the effect of “flattening” network
+performance, since all channels are used independently of their
+quality. In other words, the quality of a link experienced by
+communicating nodes is about the same as what is found by
+averaging the quality of channels. As a consequence, reliability
+and other performance indicators are less dependent on the
+quality of any single channel, making communication much
+more stable.
+Two reasonable and intuitive solutions to enhance perfor-
+mance are black listing [14] and white listing [15]. In the for-
+mer, channels with the worst performance are excluded, while
+in the latter only those channels with the better performance
+are exploited. Both solutions are based on the idea that some
+channels can be removed from (or selected for) the hopping
+sequence in order to maximize the chances that transmission
+attempts succeed.
+Over the past years, a variety of black and white listing
+techniques have been proposed in the scientific literature. A
+relevant aspect for creating a taxonomy of such techniques
+is the metric they use to identify a channel as bad or good.
+Methods based on the Received Signal Strength Indicator
+(RSSI) have been shown to be less accurate than those based
+on the Packed Delivery Ratio (PDR) [16]. Less conventional
+techniques like ETSCH rely on energy detection in idle periods
+[17], while other solutions make use of fuzzy logic [18]. It is
+also possible to use machine learning techniques to predict the
+evolution of a wireless channel, in terms of the frame delivery
+probability, given its recent past [19].
+Channel quality estimation is only one of the aspects to
+which attention has to be paid in the definition of above
+techniques. Many works recognise that link-based (local)
+listing, in which the set of good/bad channels is selected
+based on the single link, is better than global listing [20].
+The superiority of link-based approaches is reasonable, since
+mesh networks are distributed in space, and the characteristics
+of the wireless spectrum may vary noticeably even in the case
+of small movements of the nodes [21].
+Actually, a node that requires to communicate with more
+than one neighbor needs to maintain a different ν function
+for each neighbor. Selecting the correct functions can be
+done transparently, by using the information related to the
+configured (non-shared) cells in the slotframe matrix. An
+important limitation of link-based techniques is that they
+cannot be directly used for multicast/broadcast traffic.
+Black/white listing techniques also differ on the way chan-
+nel selection is shared among nodes. In particular, every time
+a technique requires a modification of the function ν (and/or
+the hopping sequence H(i)) used by a pair of nodes, both the
+transmitter NTX and the receiver NRX must agree on it. In
+the case of inconsistency between the functions νTX and νRX
+used in NTX and NRX, respectively (i.e., when νTX ̸= νRX),
+communication between the two nodes is prevented because
+it is no longer guaranteed that the channel they use for
+transmission and reception in any given slot coincide. This
+
+��
+��
+��
+��
+��
+��
+��
+��
+��
+���
+���
+��
+1j
+��
+2j
+��
+��
+3i
+4i
+4j
+5i
+5j
+��
+6i
+���
+7i
+7j
+��
+���
+8i
+8j
+���
+9i
+9j
+��
+10j
+���
+11i
+11j
+ℱ��
+����� =  � , ��
+����� =  � , −
+ℱ��
+� ACK frame
+����� =  � , ��
+����� =  � , −
+����� =  � , ��
+����� =  � , �� 
+2i
+����� =  � , ��
+����� =  � , −
+1i
+ℱ
+ℱ
+��
+Lost frame
+����� =  � , ��
+����� =  � , − 
+3j
+ℱ
+��
+ℱ���
+��
+����� =  � , ��
+����� =  � , ��� 
+6j
+ℱ���
+ℱ���
+��
+����� =  � , ���
+����� =  � , − 
+ℱ
+���
+���
+����� =  � , ���
+����� =  � , − 
+10i
+ℱ��
+Data frame with new
+hopping function ��
+ℱ data frame
+���
+���
+Double listening
+Fig. 1. Example of application of the CONSIP protocol.
+circumstance shall be absolutely avoided, because it could lead
+to the disconnection of a portion of the network.
+The ways to distribute ν to the end points of the link, and
+in particular to do so consistently, avoiding the possibility that
+νTX ̸= νRX for even just a single cell, are the main goals
+of this work. Some scientific papers treated this problem in
+a completely different way and with some assumptions [20],
+[22], for example not considering the chance that acknowl-
+edgement (ACK) frames could be lost. This simple assumption
+proved to be mostly untrue on real traffic logs we acquired on
+OpenMote B devices equipped with the OpenWSN (version
+REL-1.24.0) operating system. In fact, our measurements
+showed that the loss probability for ACK frames in our
+experimental testbed is not negligible at all, and amounts to
+about 8% in benign environmental conditions. In the same
+conditions, the loss probability for data frames was 12.6%2.
+Instead, the main idea in [23] is to leave the hopping
+function unmodified, but to lower the usage of cells associated
+with bad quality channels, which are exploited with a certain
+probability lower than one. Doing so prevents any problems
+due to inconsistency between the views of NTX and NRX,
+but at the same time limits the achievable performance. In
+fact, while this technique is suitable for reducing power
+consumption, it also causes an increase of latency.
+III. CONSISTENCY PROTOCOL
+Communication between a sender node NTX and a receiver
+node NRX in the slot characterized by ASN equal to x is only
+possible if both nodes agree on the frequency (physical chan-
+nel) to be used to send and receive data on air, respectively.
+In the case the hopping functions ν on the two sides of a link
+were unaligned, the network would quickly lose connectivity
+of some nodes, which become unreachable.
+The process of exchanging a hopping function between the
+end points of a link starts when NTX generates a new νn and
+finishes when νn is effectively in use, at which point the cells
+referring to the previous hopping function νo are switched off
+in both NTX and NRX. In the following, this process will
+be denoted, for brevity, with the term ν-exchange. Instead,
+2The experimental data on which such values were computed are included
+in the file default-101-16-15days.dat, which is downloadable from
+https://dx.doi.org/10.21227/fg62-bp39.
+CONSIP is the protocol we are proposing in this work to
+enable a ν-exchange to be performed in a consistent way.
+The main idea behind CONSIP is to have, for any cell
+Ccurr reserved for the communication between NTX and NRX,
+a backup cell Cback that is used only for the time strictly
+needed to perform a ν-exchange. After that, the cell Cback
+becomes Ccurr, i.e., the two cells reverse their role by means
+of a swap() operation. These two cells are scheduled in
+distinct slot offsets in the slotframe matrix. As a consequence,
+CONSIP does not require any modification to the hardware of
+nodes. In particular, it can be implemented in conventional de-
+vices provided with a single communication interface/antenna.
+Although there are no other constraints on the position of
+Cback within the slotframe, a reasonable choice is to interleave
+Ccurr and Cback cells so that they are evenly spaced, e.g., when
+Nslots = 101 they could be located at slot offset i and (i+50)
+mod 101. Both Ccurr and Cback are identified by a 2-tuple,
+composed of a slot offset and a hopping function. Let i and j
+be the slot offsets assigned to the two above cells, respectively.
+Then, Ccurr = (i, νo) means that the current cell in use Ccurr
+is assigned to slot offset i and that hopping function νo is used
+for transmission. Instead, Cback = (j, −) means that Cback is
+assigned to slot offset j and it cannot be used because it is
+not currently mapped to any hopping function.
+A. Simple example about protocol operation
+In Fig. 1, an example of the CONSIP operation is sketched.
+In slot 1i (i.e., slot i in slotframe 1) a frame Fνn
+x
+is sent
+to perform a ν-exchange. This frame is just a conventional
+data frame Fx that also contains the new hopping function νn
+estimated by NTX. The hopping function used to determine the
+physical channel for every slot, on either the transmitter (upper
+diagram) or the receiver side (lower diagram), is specified
+inside the box representing the slot itself. In the example, the
+first frame is lost but the retransmission of Fνn
+x
+in cell 2i
+correctly arrives to destination. For NRX, starting from slot
+2j the cell Cback = (j, νn) is activated, and the node enters
+the double listening state in which it listens on both cells Ccurr
+and Cback, using for them the hopping functions νo and νn,
+respectively. Since the frame Fνn
+x
+sent in cell 2i is confirmed
+by the related ACK frame Ax, NTX starts using exclusively
+the new hopping function νn. In particular, it performs a swap
+between the two cells by means of the swap() function, after
+
+�� ��
+���� ����� ⋀ �� �� �� �����
+����(�����, �����)
+����� =  ⋅ ,  !
+����� =  ⋅ , −
+����� =  # ,  $
+����� =  % , −
+����� =  # ,  $
+����� =  % , −
+�� ������ �����
+����� =  ⋅ ,  !
+double
+listening
+steady
+�� �� �� �����
+����� =  ⋅ , −
+�� �� �� �&'�(
+����(�����, �����)
+����� =  ⋅ , −
+�� ����)
+�� �����
+����� =  ⋅ ,  !)
+�� ����)
+�� �&'�(
+����(�����, �����)
+����� =  ⋅ ,  !)
+steady
+Transmitter Node *��
+Receiver Node *��
+{1}
+{1}
+{2}
+{3}
+{4}
+{7}
+{6}
+{5}
+Fig. 2. State machines of NTX and NRX.
+which Ccurr = (j, νn) and Cback = (i, −), i.e., the backup
+cell is deactivated. On NRX, the double listening state persists
+until it receives a frame (typically in Cback, but sometimes
+in Ccurr) whose channel is selected using the new hopping
+function νn. Only in this case, NRX can be sure that also NTX
+has switched to νn, and is consequently using Cback as current
+transmission channel. At this point NRX sets Ccurr = (j, νn)
+and Cback = (i, −). In the example of Fig. 1 this happens in
+slot 3j, which becomes the current channel.
+In the new ν-exchange with hopping function νn2 in slot
+6j, the Ax frame used to confirm Fν2n
+x
+is lost. In this case,
+node NTX continues to transmit in Ccurr = (j, νn), and only
+when Fν2n
+x
+is followed by the correct reception of Ax, which
+happens in slot 8j, node NTX can start transmitting using
+the new hopping sequence. It is worth remarking that, the
+double listening period in which NRX is active on both cells
+with both the old and new hopping sequences is mandatory
+because, after the reception of Fνn2
+x
+using νn, from the NRX
+viewpoint it is not possible to know if NTX will use νn or
+νn2 for the next transmission. In fact, if the acknowledgement
+frame Ax related to Fνn2
+x
+is correctly received by NTX, it can
+safely assume that NRX received the new hopping function
+νn2 and hence it can start using it as the current cell Ccurr =
+(i, νn2), otherwise it must assume that νn2 was not received.
+This mismatch of viewpoints between NTX and NRX may
+occur in any communications network, and it is accentuated
+in wireless networks where the probability of losses is not
+negligible at all.
+Only when the frame Fx is actually received in the cell
+related to the new hopping function νn2 (in slot 10i in the
+example), node NRX can start using Ccurr = (i, νn2) as the
+current cell, and it can disable the backup cell Cback = (j, −).
+B. Protocol description through FSMs
+To better formalize the CONSIP protocol, two finite state
+machines (FSMs) are presented in Fig. 2 for NTX and NRX. In
+particular, both FSMs start from the steady initial states with
+Ccurr = (i, νo) and Cback = (j, −) (see the arc labeled {1}
+in the figure). These initial steady states represent the normal
+operating condition of TSCH, in which only one cell is active
+and there are no ongoing updates of ν.
+Regarding arc {2} of the NTX FSM, each time a frame
+delivering a new hopping function Fνn
+x
+is acknowledged
+in Ccurr, the node NTX can set the cell Ccurr to νn. For
+doing so, the node swaps the two cells (i.e., it invokes
+swap(Ccurr, Cback)), and then it sets the hopping function
+νn in the current cell and disables the hopping function, and
+consequently the ability to transmit, in the backup cell. This
+is performed by means of the two operations Ccurr = (·, νn)
+and Cback = (·, −), where the symbol “·” means that the slot
+offset is not changed.
+Regarding the FSM of NRX, it consists of two states.
+Each time a frame containing a new hopping function νn is
+received in Ccurr, the backup cell is activated and the FSM
+enters the double listening state through arc {3}, in which
+NRX receives on both cells, consequently increasing its power
+consumption. The arc labelled {4} either corresponds to the
+NTX’s intention to change on the fly the previously transferred
+hopping function νn with a new hopping function νn′ or, more
+typically, it is due to a retransmission of the previous frames
+for which the related ACK did not arrive to destination. In the
+latter case νn′ = νn.
+The arrival of a frame in the backup cell, as for arc {5},
+confirms to NRX that the transmitter is correctly using the
+new hopping function. As a consequence, the receiving node
+can start using the backup cell as current cell, by invoking
+swap(Ccurr, Cback), after which it can disable the backup
+cell Cback = (·, −).
+A frame Fνn′
+x
+delivering a new hopping function νn′ that
+is received in the backup channel, as depicted for arc {6},
+has simultaneously two consequences. The first is to confirm
+to NRX that NTX is using the new hopping function νn, and
+the second to communicate that NTX wants to perform a new
+ν-exchange using νn′. For this reason the use of the swap()
+function is required.
+Finally, arc {7} accounts for the unlikely case that NTX
+decides to abort the ν-exchange, and it continues transmitting
+normal frames in Ccurr. For instance, this may happen if the
+sender node detects that the quality of channels has changed
+and the current hopping function νo is again the best one.
+IV. EXPERIMENTAL SETUP
+A discrete event simulator named TSCH-predictor, which
+was developed within the SimPy framework, was used to
+evaluate the effectiveness of CONSIP. Differently from other
+more common simulators such as TSCH-Sim [24] and 6TiSCH
+[25], TSCH-predictor has the advantage to be noticeably
+simpler, and consequently it permits to easily implement and
+
+evaluate new algorithms and techniques based on TSCH.
+TSCH-predictor was profitably used in other research works
+like [26], [27], and more information about its features can be
+found in [27].
+The main settings used in all the experimental campaigns
+were selected as follows: Nslots = 101, slot duration 20 ms,
+Nch = 16, frame loss probability ϵf = 12.6%, and ACK
+loss probability ϵa = 8.0%. In the experiments, both proba-
+bilities ϵf and ϵa were left constant over time. This is not a
+big limitation, as the sensitivity of CONSIP with respect to
+channel errors is not the main focus of this work. However,
+such analysis may be of interest, and is left as future work.
+As specified, these two values were derived from a real
+experimental testbed deployed in our lab. Finally, the duration
+of each experiment was set to 10 years, which ensures for
+results good statistical significance.
+The size of each packet sent in the simulation is Ltot =
+Lheader +Lpayload +LIE, where Lheader = 29 B is the overall
+size of both the PHY and MAC headers, Lpayload = 30 B
+refers to the payload (we chose a relatively small size for it,
+as happens in typical industrial networks and WSANs), and
+LIE concerns the information element (IE). In particular, in
+this work the IE is used to encode and transfer ν. The IE,
+whose size is LIE = LIEh + LIEp, is a specific configurable
+attribute that the IEEE 802.15.4 standards permits to attach
+to a frame, and consists of an IE header LIEh = 2 B and an
+IE payload LIEp that is a configuration parameter and must
+have enough room to store ν. In the following, the value LIEp
+was fixed to 16 B unless explicitly stated. The exact way the
+ν function is encoded depends on the specific implementation
+of the black/white listing technique, and is outside the scope
+of this work.
+In this new version of the simulator we exploited the
+energy model described in [28]. In particular, the energy to
+transmit a data frame is Etx = Etx0 + etx · Ltot, and the
+energy to receive a data frame is Erx = Erx0 + erx · Ltot,
+where Etx0 = 7 µJ, etx = 2 µJ/B, Erx0 = 65 µJ, and
+erx = 1.3 µJ/B. Instead, the energy to transmit an ACK frame
+(whose size is 33 B) is EACK
+tx
+= 106 µJ, the energy to receive
+it is EACK
+rx
+= 79 µJ, and the energy spent for idle listening,
+when the receiver switches its interface on without receiving
+any data, is Elisten = 138 µJ.
+V. RESULTS
+The effectiveness of CONSIP was assessed through an
+extensive experimental campaign aimed at analyzing it from
+the point of view of two relevant key performance indicators,
+namely, power consumption and ν-exchange latency. There
+is no need to analyze it also from the point of view of
+reliability, because in CONSIP every packet is guaranteed the
+same number of retries (upon errors) as the unmodified TSCH.
+A. Power consumption
+The first set of experiments is targeted at analyzing the
+amount of energy used by CONSIP when a ν-exchange is
+performed cyclically with a period equal to Tupdate. Several
+values are chosen for this parameter, in the range from some
+minutes to a few hours. Two separate periods were set for the
+traffic flow between NTX and NRX, that is, Tapp = 30 s and
+Tapp = 5 s. The former value mimics a reasonable generation
+period for sensors located in leaf nodes. Instead, the latter
+value (i.e., Tapp = 5 s) models the links between nodes near
+the root, in which the aggregation of the traffic generated
+from the lower layers of the topology increases the amount of
+cells that are actually used for transmission. Selecting periodic
+transmission patterns does not limit in any way the validity of
+the proposed method.
+Table I compares network behavior for different values of
+Tupdate (and Tapp) with respect to the case when CONSIP is
+disabled. Comparison with conventional TSCH operation (i.e.,
+when CONSIP is disabled) is quite relevant, because it permits
+to statistically detect possible drawbacks of this method. We
+did not find in the scientific literature any methods similar to
+CONSIP to perform a meaningful comparison.
+Starting from Tupdate and Tapp, the number of samples
+for each channel that can be exploited to compute νn is
+#ν =
+Tupdate·60
+Tapp·16
+(the update time is expressed in minutes).
+This value just provides an indication about the amount of new
+information that can be exploited, on average, by a black/white
+listing algorithm to compute νn. However, it is not related to
+the way this computation is actually performed, and not even
+to the quality of the channel. More important, it is not relevant
+to CONSIP. The value of #ν is reported in the related column
+of the table. When Tapp = 30 s, in order to have at least one
+sample per channel on every update of the hopping function
+(that is, #ν ≥ 1) the update period Tupdate must be set to a
+value greater than 8 min.
+Regarding NTX, all the energy related to communication
+is spent for data transmission. The reason why P NTX
+tx/tot is
+inversely proportional to Tupdate is that, every time a ν-
+exchange is triggered, the size of the packet to be transmitted
+is increased by LIE bytes.
+Regarding power consumption on NRX, there is a sensible
+increase in the energy P NRX
+listen that is wasted because of idle
+listening. This is due to the fact that when NRX is in the
+double listening state, it enables its receiving interface in both
+cell Ccurr and Cback, but one of the two cells remains unused
+because the sender node NTX transmits only in one cell, either
+Ccurr or Cback. In particular, the growth in both the total power
+consumption Ptot and the power consumption on NRX that is
+observed when CONSIP is employed mostly depends on the
+increase of Plisten, i.e., to the higher amount of idle listening
+because of the aforementioned double listening.
+Analyzing the case Tapp = 30 s, when #ν ≃ 1 a new νn
+can be obtained exploiting, on average, about one additional
+sample for each channel. This can be likely considered a
+worst condition from the point of view of energy, and the
+relative increase of the total power consumption (Ptot) is
+equal to +5.80 %, which is a reasonable value for many
+application contexts. When the ν-exchange is performed at a
+slower pace (i.e., every Tupdate = 30 min and 60 min, which
+means #ν = 3.75 and #ν = 7.5) the relative increase in
+
+TABLE I
+POWER CONSUMPTION AND LATENCY VS. DIFFERENT EXCHANGE PERIODS OF ν.
+Power consumption
+Latency
+Listing
+Tupdate
+#ν
+P NTX
+tx/tot
+P NRX
+rx
+P NRX
+listen
+P NRX
+tot
+Ptot
+µd
+σd
+dp99
+dp99.9
+dmax
+[min]
+[µW]
+[µW]
+[µW]
+[%]
+[s]
+Disabled
+-
+-
+8.622
+9.823
+62.596
+72.419
+81.041
+-
+1.311
+1.006
+4.900
+7.200
+9.460
+Enabled
+Tapp = 30 s
+7.5
+0.94
+8.711
+9.880
+67.151
+77.031
+85.742
++5.80%
+1.311
+1.006
+4.900
+7.200
+9.480
+15
+1.88
+8.667
+9.851
+64.873
+74.725
+83.391
++2.90%
+1.311
+1.006
+4.900
+7.200
+9.460
+30
+3.75
+8.645
+9.837
+63.735
+73.572
+82.216
++1.45%
+1.311
+1.006
+4.900
+7.200
+9.440
+60
+7.5
+8.633
+9.830
+63.166
+72.995
+81.629
++0.73%
+1.311
+1.006
+4.900
+7.220
+9.480
+120
+15
+8.628
+9.826
+62.881
+72.707
+81.335
++0.36%
+1.311
+1.006
+4.900
+7.200
+9.460
+240
+30
+8.625
+9.824
+62.739
+72.563
+81.188
++0.18%
+1.311
+1.006
+4.900
+7.220
+9.480
+Disabled
+-
+-
+51.732
+58.932
+33.995
+92.927
+144.659
+-
+1.384
+1.091
+5.340
+7.880
+21.660
+Enabled
+Tapp = 5 s
+7.5
+5.63
+51.820
+58.990
+34.748
+93.737
+145.557
++0.621%
+1.383
+1.090
+5.340
+7.880
+20.640
+15
+11.25
+51.776
+58.961
+34.371
+93.332
+145.108
++0.311%
+1.383
+1.091
+5.340
+7.880
+21.660
+30
+22.5
+51.754
+58.946
+34.183
+93.130
+144.883
++0.155%
+1.384
+1.091
+5.360
+7.880
+21.660
+60
+45
+51.743
+58.939
+34.089
+93.028
+144.771
++0.078%
+1.384
+1.091
+5.340
+7.880
+20.640
+120
+90
+51.737
+58.936
+34.042
+92.978
+144.715
++0.039%
+1.384
+1.091
+5.340
+7.880
+21.660
+240
+180
+51.734
+58.934
+34.019
+92.952
+144.687
++0.019%
+1.384
+1.091
+5.360
+7.880
+21.660
+terms of total power consumption is almost negligible and
+equal to +1.45 % and +0.73 %, respectively. This confirms
+that the main drawback of CONSIP is not the additional
+energy consumption, but the need to allocate twice as much the
+number of cells for each link if compared with a scheduling
+strategy without CONSIP. The results with Tapp
+= 5 s,
+in which case the maximum relative increase in the total
+power consumption is only +0.621 %, further corroborate this
+conclusion. However, this limitation is typically problematic
+only for a small subset of network topologies, characterized
+by a larger number of nodes and a high density.
+The last set of columns in Table I reports some performance
+indicators related to latency. They show that the influence of
+CONSIP on latency is irrelevant. The only statistical indices
+that are not the same for all the experimental conditions are
+percentiles (dp99.9 for Tapp = 30 s, and dp99 for Tapp = 5 s)
+and the maximum value (dmax), but they are anyway very
+similar. This behaviour is somehow expected, since high-order
+percentiles and the maximum converge to their real values
+much more slowly than other statistical indices such as the
+mean value and lower-order percentiles.
+TABLE II
+POWER CONSUMPTION WITH Tapp = 30 s AND T update = 30 min VS.
+DIFFERENT SIZES OF ν.
+LIEp
+P NTX
+tx/tot
+P NRX
+rx
+P NRX
+listen
+P NRX
+tot
+Ptot
+[B]
+[µW]
+[µW]
+[µW]
+[%]
+16
+8.645
+9.837
+63.735
+73.572
+82.216
++1.450%
+14
+8.642
+9.835
+63.735
+73.570
+82.212
++1.445%
+12
+8.639
+9.833
+63.735
+73.568
+82.207
++1.439%
+10
+8.636
+9.832
+63.735
+73.567
+82.203
++1.433%
+8
+8.633
+9.830
+63.735
+73.565
+82.198
++1.428%
+Table II shows the results of another experimental campaign
+aimed at analyzing the effect on power consumption of the
+size LIEp of the encoding of ν. The values Tapp = 30 s
+and Tupdate = 30 min are representative of typical operating
+conditions, and therefore they have been left unmodified.
+As highlighted in the rightmost columns of the result table,
+the relative increase of Ptot ranges from +1.428 % (when
+LIEp = 8 B) to +1.450 % (when LIEp = 16 B), which means
+that advanced optimizations on the way ν is encoded, that
+lead to a further reduction of LIEp, only lead to insignificant
+improvements from the point of view of energy consumption.
+However, its reduction is important because it permits to
+increase the room available for the payload in the frame.
+An additional experimental campaign was carried out, again
+with Tapp = 30 s and Tupdate = 30 min, to analyze the effect
+of the placement of the two cells, Ccurr and Cback. Two cases
+were considered, where the cells were equally spaced (in the
+slot offsets 1 and 51, respectively) and contiguous (in the slot
+offsets 1 and 2, respectively). As expected, we verified that
+these is no influence on power consumption, and the influence
+on latency was irrelevant. For instance, the average latency was
+the same in the two cases, while standard deviation passes
+from σd = 1005.81 ms in the case of equally spaced cells to
+σd = 1006.07 ms in the case of contiguous cells.
+B. Update latency
+Another important performance indicator is the time to
+complete a ν-exchange. In fact, at the end of the whole
+process, when NRX returns in the steady state, only one of
+the two cells is effectively used to transmit, and from that
+point on the system behaves as standard TSCH, with the only
+exception that Cback remains reserved for future exchanges,
+although not in use.
+Referring to Fig. 3, to analyze timings four points in time
+were identified, in correspondence to the main events that
+
+��
+��
+��
+��
+��
+��
+��
+1j
+��
+2j
+��
+��
+3i
+��
+4i
+4j
+5i
+5j
+��
+ℱ��
+2i
+1i
+ℱ
+��
+3j
+ℱ
+��
+���
+���
+��
+ℱ��
+t
+���
+���
+���
+��
+Fig. 3. Main temporal events involved in a ν-exchange in CONSIP.
+make up a ν-exchange. Timestamps on these specific events
+are acquired with the resolution of the ASN (one slot time),
+which in this experimental campaign corresponds to 20 ms. In
+particular, they are:
+1) tUR (update request) is the time when a ν-exchange is
+started.
+2) tDL (double listening) represents the time when a new
+hopping function νn is received by NRX. At this time
+the receiver enters the double listening state, in which it
+hears from both cells Ccurr and Cback.
+3) tSW (swap) is the time when NTX starts transmitting
+using the new hopping function νn.
+4) tE (end) is the time when NRX starts receiving only
+using the new hopping function νn, consequently exit-
+ing the double listening state. This ends the whole ν-
+exchange process.
+Starting from these four timestamps, we analyzed three main
+kinds of latency, namely:
+• dSW = tSW − tUR (swap latency) is the time elapsing
+from the beginning of a ν-exchange to the time when the
+new hopping function νn is actually used to transmit data
+from NTX to NRX.
+• dDL = tE − tDL (double listening latency) is the time
+interval for which NRX remains in the double listening
+state. This interval is characterized by a higher amount
+of energy consumption.
+• dtot = tE − tUR (total latency) is the time needed to
+complete the whole ν-exchange process, starting from the
+request and up to the update of the hopping function.
+A further experiment was carried out where Tapp = 30 s and
+Tupdate = 30 min. The main statistic indicators we obtained
+for the above latencies are reported in Table III. Regarding
+the swap latency dSW, its average is 1.491 s. This value can
+be explained by analyzing the plot of the Probability Density
+Function (PDF) in Fig. 4. The plot shows a step function, and
+TABLE III
+LATENCY RELATED TO A ν-EXCHANGE IN CONSIP.
+Latency
+µd
+σd
+dmin
+dp99
+dp99.9
+dmax
+[s]
+dSW = tSW − tUR
+1.491
+1.256
+0.0
+5.880
+9.080
+14.100
+dDL = tE − tDL
+30.005
+1.511
+19.180
+33.340
+35.360
+41.420
+dtot = tE − tUR
+31.294
+1.009
+30.000
+34.900
+37.200
+41.660
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+0
+2
+4
+6
+8
+10
+dSW [s]
+PDF
+CDF
+Fig. 4. PDF and CDF of dSW.
+the width of each step is equal to 2.02 s, which corresponds to
+the slotframe duration. Within each step, latency is uniformly
+distributed because periods at the application layer (Tupdate =
+30 min) and at the MAC layer (that is, the repetition period of
+a cell in the slotframe, equal to 2.02 s) are prime numbers and
+the two processes can be treated as they were independent.
+The first (and higher) step is related to the frames that arrived
+to NRX after exactly one transmission attempt, the second
+step refers to frames transmitted twice, and so on. The same
+plot also reports the Cumulative Distribution Function (CDF)
+of the swap latency dSW, which can be used to determine
+the expected number of ν-exchange that experienced a swap
+latency smaller than a given value. For the same quantity
+dSW, the maximum latency is dmax = 14.100 s, which refers
+to a frame that was transmitted 7 times (i.e., ⌈ dmax
+2.02 ⌉) before
+reaching the destination.
+Regarding the double listening latency dDL, its average
+value is about 30 s, which is equal to Tapp. This is unsur-
+prising, because only when a frame is transmitted in Cback
+(see arc {5} of the NRX state machine in Fig. 2), the new
+hopping function νn is definitely activated, and the double
+listening phase ends. Since packets are generated cyclically
+with period Tapp = 30 s, excluding retransmissions, this frame
+typically arrives 30 s after the frame containing νn. This is
+one of the drawbacks of CONSIP: in other words, the dDL
+interval, in which a considerable amount of energy is wasted
+due to idle listening because both cells are active, depends
+on the link usage. However, in those links that experience
+higher traffic, that typically correspond to the levels in the tree
+network topology closest to the root node, where black listing
+techniques are more important to reduce the overall number of
+retransmissions, the interval between two successive packets
+is usually shorter than in the links close to the leaf nodes.
+An important property is that, at least two frames are need
+to perform the whole ν-exchange. This is confirmed by the
+minimum value reported in the row of the table that refers to
+the total latency dtot.
+VI. CONCLUSIONS
+In the context of black/white listing techniques, a crucial
+point is how to spread the information about the channels
+to be used for the transmission between nodes. When the
+
+hopping sequence is defined on a per-link basis, only the
+two end points are involved. In the case of inconsistency,
+communication in the network can be prevented, with possible
+definitive disconnections of subsets of nodes.
+The CONSIP technique was proposed to counteract this
+problem by means of a backup cell. Each time a modification
+is triggered about the channels to be used for transmission over
+the link connecting two nodes, for a limited period of time both
+cells are exploited for communication. Doing so guarantees
+that the information seen by the two involved nodes is always
+updated in a coherent way, irrespective of the number of trans-
+mission errors that affected either data or acknowledgement
+frames. The experimental analysis of CONSIP, performed by
+means of a simulator that was configured with data derived
+from a real setup, highlights its effectiveness. In particular,
+CONSIP does not affect communication latency, and has a
+small impact on energy consumption. Its main drawback is
+the need to reserve a backup cell per link, which can limit the
+number of nodes in dense networks.
+Future works include improvements related to the CONSIP
+technique, which are aimed for instance to reduce energy
+consumption further, or to lower the number of backup cells
+that need to be reserved by the protocol. Future directions
+include the usage of CONSIP in the implementation of a
+black/white listing technique.
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+

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+Draft version February 1, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Detection of a high-velocity prominence eruption leading to a CME associated with a superflare
+on the RS CVn-type star V1355 Orionis
+Shun Inoue
+,1 Hiroyuki Maehara
+,2 Yuta Notsu
+,3, 4, 5 Kosuke Namekata
+,6 Satoshi Honda
+,7
+Keiichi Namizaki,8 Daisaku Nogami,8, 9 and Kazunari Shibata10, 11
+1 Department of Physics, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan
+2Okayama Branch Office, Subaru Telescope, NAOJ, NINS, Kamogata, Asakuchi, Okayama, 719-0232, Japan
+3Laboratory for Atmospheric and Space Physics, University of Colorado Boulder, 3665 Discovery Drive, Boulder, CO 80303, USA
+4 National Solar Observatory, 3665 Discovery Drive, Boulder, CO 80303, USA
+5Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, 152-8551, Japan
+6ALMA Project, NAOJ, NINS, Osawa, Mitaka, Tokyo, 181-8588, Japan
+7Nishi-Harima Astronomical Observatory, Center for Astronomy, University of Hyogo, Sayo, Hyogo, 679-5313, Japan
+8Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan
+9Astronomical Observatory, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan
+10Kwasan Observatory, Kyoto University, Yamashina, Kyoto, 607-8471, Japan
+11 School of Science and Engineering, Doshisha University, Kyotanabe, Kyoto, 610-0321, Japan
+ABSTRACT
+Stellar coronal mass ejections (CMEs) have recently received much attention for their impacts on
+exoplanets and stellar evolution. Detecting prominence eruptions, the initial phase of CMEs, as the
+blue-shifted excess component of Balmer lines is a technique to capture stellar CMEs. However, most
+of prominence eruptions identified thus far have been slow and less than the surface escape velocity.
+Therefore, whether these eruptions were developing into CMEs remained unknown. In this study, we
+conducted simultaneous optical photometric observations with Transiting Exoplanet Survey Satellite
+and optical spectroscopic observations with the 3.8m Seimei Telescope for the RS CVn-type star
+V1355 Orionis that frequently produces large-scale superflares.
+We detected a superflare releasing
+7.0 × 1035 erg. In the early stage of this flare, a blue-shifted excess component of Hα extending its
+velocity up to 760 − 1690 km s−1 was observed and thought to originate from prominence eruptions.
+The velocity greatly exceeds the escape velocity (i.e., ∼ 350km s−1), which provides important evidence
+that stellar prominence eruptions can develop into CMEs. Furthermore, we found that the prominence
+is very massive (9.5 × 1018 g < M < 1.4 × 1021 g). These data will clarify whether such events follow
+existing theories and scaling laws on solar flares and CMEs even when the energy scale far exceeds
+solar cases.
+Keywords: stars: activity — stars: flare —stars: individual (V1355 Orionis) — stars: mass-loss
+1. INTRODUCTION
+Solar flares are explosive phenomena wherein mag-
+netic energy stored around sunspots is suddenly re-
+leased through magnetic reconnection (e.g., Shibata &
+Magara 2011).
+Generally, a solar flare releases about
+1026 −1032 erg. Emission in a wide range of wavelengths
+[email protected]
+from radio waves to X-rays occurs during a flare. Gen-
+erally, prominence eruptions on the Sun, which are as-
+sociated with flares (e.g., Shinha et al. 2019), are ob-
+served as Hα emission and Hα absorption when they
+erupt outside a limb and on a disk, respectively (Par-
+enti 2014; Otsu et al. 2022).
+Prominence or filament
+eruptions can lead to coronal mass ejections (CMEs)
+when the prominence velocity is sufficiently large (e.g.,
+Gopalswamy et al. 2003; Shibata & Magara 2011).
+Flares are widely observed both on the Sun and
+stars. In the case of stars, “superflares,” which release
+10 times larger energies than the largest solar flares,
+arXiv:2301.13453v1  [astro-ph.SR]  31 Jan 2023
+
+ID2
+Inoue et al.
+Table 1. Basic physical parameters of the K-type subgiant star of the binary
+Spectral Type
+V ∗
+V − RC∗
+d
+† Porb ‡
+Prot $
+R
+§
+L
+¶
+M
+♮ Teff ♯
+ve
+⋆
+(mag)
+(mag)
+(pc)
+(days)
+(days)
+(R⊙)
+(L⊙)
+(M⊙)
+(K)
+(km/s)
+(1)
+(2)
+(3)
+(4)
+(5)
+(6)
+(7)
+(8)
+(9)
+(10)
+(11)
+K0-2IV
+8.98
+0.53
+127.4
+3.87
+3.86
+4.1
+6.4
+1.3
+4750
+-347
+*The V -band magnitude and difference between V and and RC-bands.
+† Stellar distance.
+‡ Orbital period.
+$Rotation period.
+§ Radius.
+¶Luminosity.
+♮ Mass.
+♯ Effective temperature.
+⋆Escape velocity at the surface.
+References—(1),(5)-(11):Strassmeier (2000), (2), (3):Cutispoto et al. (1995)
+, (4):Gaia Collaboration et al. (2016)
+have been confirmed (e.g., Maehara et al. 2012). Spec-
+troscopic observations of stellar flares sometimes show
+“blue shifts” (or “blue asymmetries”) wherein chromo-
+spheric lines during flares are not symmetric, but are
+enhanced only at shorter wavelengths (e.g., Houdebine
+et al. 1990; Gunn et al. 1994; Fuhrmeister & Schmitt
+2004; Fuhrmeister et al. 2008, 2011; Vida et al. 2016;
+Honda et al. 2018; Vida et al. 2019; Muheki et al. 2020;
+Maehara et al. 2021).
+An excess component with a
+shorter wavelength than the rest line center is observed,
+indicating that the source is flying toward us, consid-
+ering the Doppler effect. Therefore, blue shifts might
+suggest that an upward-moving plasma exists, such as
+prominence eruptions (Otsu et al. 2022) or a chromo-
+spheric temperature (cool) upflow associated with chro-
+mospheric evaporation (Tei et al. 2018). One technique
+to determine whether a blue-shifted excess component
+originates from cool upflows or prominence eruptions is
+to estimate it based on its Doppler velocity. Typically,
+cool upflows have a velocity of ∼ 100 km s−1 in solar
+flares (e.g., Kennedy et al. 2015).
+Therefore, we can
+expect that blue shifts that significantly exceed this ve-
+locity originate from prominence eruptions.
+Most of the blue shifts discovered thus far have been
+relatively slow (< 500km s−1). Almost no solid evidence
+exists that they originated from prominence eruptions
+and these eruptions triggered a CME. Namekata et al.
+(2022a) first discovered solid evidence of a stellar fila-
+ment eruption by detecting the blue-shifted absorption
+component of Hα. By using the length scale and the
+velocity of the ejected plasma, Namekata et al. (2022a)
+confirmed that the erupted filament very likely devel-
+oped into CMEs.
+Further, few cases of blue shifts in
+flares larger than 1035 erg emerge. Generally, a positive
+correlation exists between the energy of solar flares and
+those of associated prominence eruptions (e.g., Aarnio
+et al. 2011; Takahashi et al. 2016). Previous blue shift
+examples found that events on stars generally come as an
+extension of this correlation (e.g., Moschou et al. 2019).
+Therefore, if we can detect a blue shift during a par-
+ticularly large-scale superflare (> 1035 erg), its kinetic
+energy is likely to be excessively large. This becomes
+reliable evidence of a stellar CME. Moreover, detection
+of these events can confirm whether the relation between
+flare energy and the size of the plasma eruption could
+be extended to this region.
+RS CVn-type stars are magnetically active and have
+been observed to produce large superflares (Tsuboi et al.
+2016, and references therein). Many previous studies of
+flares on RS CVn stars have been conducted in X-ray,
+whereas only few studies have been conducted with op-
+tical spectroscopic observations. Particulaly, large-scale
+superflares (i.e., > 1035erg) frequently occur on RS CVn
+stars. In recent years, stellar CMEs have attracted much
+
+A high-velocity prominence eruption on V1355 Orionis
+3
+Figure 1. White-light light curves of V1355 Orionis observed with TESS. (a) Long-term light curve of V1355 Orionis for
+BJD=2459201.7-2459227.5. The vertical axis represents the flux normalized by the median value. The vertical light blue line
+and the horizontal green bars indicate the event discussed in this paper and the period of monitoring observations by Seimei
+telescope/KOOLS-IFU, respectively. (b) Enlarged light curve around December 19/BJD=2459203.11297. The horizontal axis
+represents the time (unit of minutes) from BJD=2459203.11297. The skyblue dashed line shows the global trend of the stellar
+rotational modulation fitted for −500 − −10 min and 150 − 380 min with a polynomial equation.
+attention for their impacts on mass/angular-momentum
+loss and the environment of surrounding exoplanets.
+If existing correlations between prominence eruptions
+and flare energies hold for particularly large superflares
+(> 1035erg), prominence eruptions that accompany such
+flares will lead to particularly large-scale CMEs. Con-
+sequently, they will have particularly large effects on
+stellar evolution and exoplanets (Osten & Wolk 2015;
+Airapetian et al. 2020). Therefore, optical spectroscopic
+observations of RS CVn-type stars are important be-
+cause they enable the detection of particularly large-
+scale superflares, which are quite infrequent to be ob-
+served on normal main-sequence stars, with only a short
+period of monitored observations.
+In this study, optical spectroscopic observations by
+the 3.8m Seimei Telescope (Kurita et al. 2020) and pho-
+tometric observations by Transiting Exoplanet Survey
+Satellite (TESS; Ricker et al. 2015) were simultaneously
+performed to V1355 Orionis, a RS CVn-type star. The
+observations and analysis (Section 2), results (Section
+3), and discussion (Section 4) on the details of the su-
+perflare and the associated blue shift obtained through
+the simultaneous observations are reported in this pa-
+per.
+2. OBSERVATIONS AND ANALYSES
+2.1. Target star : V1355 Orionis
+V1355 Orionis (=HD291095) is an RS CVn type bi-
+nary system discovered through the ROSAT WFC all-
+sky X-ray survey (Pounds et al. 1993; Pye et al. 1995).
+V1355 Orionis has been investigated by Cutispoto et al.
+(1995), Osten & Saar (1998) and Strassmeier (2000).
+These studies have shown that this binary comprises
+K0-2IV and G1V stars. Table 1 summarizes the basic
+physical parameters of the K-type subgiant star. Strass-
+meier (2000) reported a large flare on V1355 Orionis
+in April 1998, which showed 70 times larger equivalent
+width of Hα than the pre-flare.
+2.2. Simultaneous observations
+2.2.1. Photometric observation : TESS
+Transiting Exoplanet Survey Satellite (TESS) ob-
+served V1355 Orionis in Sector 34 for 27 days with the
+2 min time cadence. Figure 1 (a) shows the TESS light
+curve for this period (BJD=2459201.7-2459227.5). Fig-
+ure 1 (b) shows the enlarged TESS light curve around
+the flare described in this paper. The quiescent radia-
+tion component during the flare is estimated by fitting
+a polynomial to the light curve for −500 − −10 min and
++150 − +380 min from the flare start (see the skyblue
+dashed line in Figure 1 (b)). Figure 2 (a) shows the light
+curve for the flare component, created by subtracting
+the quiescent component from the light curve of TESS
+during the flare. We calculated the bolometric energy
+of the flare from this detrended light curve following the
+method of Shibayama et al. (2013). See Section 3.1 for
+more details about caluculating the flare energy.
+
+(a)
+December 19 event
+(b)
+Global Trend
+1.10
+1.03
+1.08
+Media
+1.06
+by
+1.01
+ Normalized
+1.04
+-150
+-75
+75
+150
+0
+225
+300
+Time [min]
+1.02
+xnl
+1.00
+0.98
+Seimei/KOOLS-IFU
+2203
+2208
+2218
+2223
+2228
+2213
+Time from BJD=2457000 [day]4
+Inoue et al.
+Figure 2. Light curves of V1355 Orionis during the December 19 event. (a) Detrended light curve of the TESS white light with
+background removed. This corresponds to the TESS normalized flux (black line in Figure 1 (b)) subtracted by the rotational
+modulation (skyblue line in Figure 1(b)). The black dotted line represents the zero level. The orange range indicates the time
+period determined as the “pre-flare” in our spectral analysis. (b) Light curve of the Hα for the same time period as in panel
+(a). The vertical axis shows the equivalent width of Hα. The black dotted line represents the level of background, calculated
+as the average of the equivalent width of the pre-flare. Note that in this light curve, the equivalent width is negative for the
+emission line flux.
+2.2.2. Spectroscopic observation : the Seimei telescope
+We observed V1355 Orionis using KOOLS-IFU (Ky-
+oto Okayama Optical Low-dispersion Spectrograph with
+optical-fiber Integral Field Unit; Matsubayashi et al.
+(2019)), the spectrograph onboard the 3.8m Seimei tele-
+scope (Kurita et al. 2020). KOOLS-IFU covers 5800-
+8000 ˚A with wavelength resolution R (= λ/∆λ) of
+∼ 2000. Therefore, it can capture the Hα line.
+Observations were made over eight nights in conjunc-
+tion with Sector 34 of TESS. The flare investigated in
+this study occurred on the first day of our observations.
+The detailed observation periods are indicated by the
+horizontal green bars in Figure 1 (a). The exposure time
+was set to 60 s on all observation dates. The signal-to-
+noise ratio (S/N) at the continuum around Hα line was
+more than 10.
+Data processing was conducted in the same manner as
+that of Namekata et al. (2020, 2022a,b) with IRAF pack-
+age, Pyraf software, and the data reduction packages
+developed by Matsubayashi et al. (2019). Using the Hα
+line profile at each time frame, we investigated the time
+variation of the equivalent width during the flare. Fig-
+ure 2 (b) shows the light curve of the equivalent width
+of Hα. Further, the equivalent width was calculated, af-
+ter normalizing Hα emission by the nearby continuum
+level, integrating it for 6518 − 6582 ˚A. The reason why
+the range of integration is asymmetric to the line cen-
+ter of Hα (= 6562.8 ˚A) is due to the blue-shifted excess
+component. See Section 3.2 for more information about
+the blue-shifted excess component.
+3. RESULTS
+3.1. White light flare energy
+As shown in Figure 2 (a), the white light flare lasted
+about 110 min. By setting the flux of Figure 2 (a) to
+C′
+flare (= luminosity of flare/luminosity of star) in the
+equation (5) of Shibayama et al. (2013), we estimated
+the flare area Aflare as follows,
+Aflare(t) =
+πC′
+flare(t) �
+i=1,2
+�
+R2
+i
+�
+RλBλ(Ti)dλ
+�
+�
+RλBλ(Tflare)dλ
+(1)
+where λ is the wavelength, Bλ is the Planck function,
+Rλ is the TESS response function (Ricker et al. 2015),
+and Ti is the effective temperature of the K- and G-type
+stars of the binary (4750 K and 5780 K, respectively;
+Strassmeier 2000). Further, Tflare is the flare temper-
+ature of 10000 K (Mochnacki & Zirin 1980; Hawley &
+Fisher 1992), and Ri is the radius of the K- and G-
+type stars of the binary (4.1R⊙ and 1.0R⊙, respectively;
+Strassmeier 2000). Assuming that the flare can be ap-
+proximated by blackbody radiation with a temperature
+of Tflare = 10000 K, flare luminosity Lflare is
+Lflare(t) = σSBT 4
+flareAflare(t)
+(2)
+where σSB is the Stefan-Boltzmann constant. Finally,
+by integrating Lflare with the duration of the white light
+
+Pre-flare
+xnl
+(a)
+White Light
+0.02
+Detrended
+3.5
+(b)
+Hα
+-2.0
+Ei
+-0.5
+100
+100
+200
+0A high-velocity prominence eruption on V1355 Orionis
+5
+Figure 3. Time variation of the Hα spectrum in the early stages of the flare. (a) Line profile of Hα emission. Each spectrum
+is a composite of three (= 3 min) spectra. The bottom and top axes are the wavelength and Doppler velocity from the line
+center, respectively. The intensity of each spectrum is normalized by continuum. The vertical dotted and horizontal dashed
+lines represent the line center of Hα and the continuum level for each spectrum, respectively. The spectrum at each time is
+vertically shifted, and the time shown in the upper right corner of each spectrum represents the time from the flare start. The
+red dash-dot lines represent the spectra at the pre-flare (−15 min to 0 min). (b) Pre-flare subtracted spectrum displayed in the
+same manner as in panel (a). Horizontal dashed line represents the zero level for each spectrum. The black minus red in panel
+(a) denotes the spectrum at each time.
+
+Velocity [km/s]
+Velocity [km/s]
+(a)
+Pre-flare
+(b)
+Each Time
+Each Time
+45-48 mir
+45-48 min
+4.24
+42-44 min
+3.97/
+38-40 min
+38-40 min
+3.70
+tant
+34-37 min
+3.43
+0.99
+Cons
+Constar
+31-33 min
+31-33 min
++
+0.88
+Intensity
++
+27-29 min
+Intensi
+ted
+3-26 min
+23-26 min
+0.66
+Subtra
+20-22 min
+2.
+JON
+16-18 min
+16-18 min
+2.08
+0.44
+12-15 min
+1.81
+0.33
+9-11 min
+-11 min
+1.54
+0.22F
+5-7 min
+5-7 min
+1.27
+0.11
+min
+1.00
+0.00k
+Wavelength [A]
+Wavelength [A]6
+Inoue et al.
+flare (∼ 110 min), the bolometric flare energy Ebol is
+Ebol =
+�
+Lflare(t) dt = 7.0 × 1035 erg.
+(3)
+Given that the K-type star is more magnetically active
+and the bolometric flare energy is very large, we consider
+this flare to have occurred on the K-type star of the
+binary.
+3.2. Hα line profile during the flare
+Our spectroscopic observation revealed that a remark-
+able blue-shifted excess component exists in the Hα
+emission line for ∼ 30 min after the start of the flare.
+Figure 3 (a) shows the spectra during the time when
+the blue-shifted excess component was observed during
+the flare. We estimated the spectrum of the quiescent
+component, denoted by red spectrum in Figure 3 (a), by
+combining the spectra from 15 min before the flare start
+(“pre-flare”). Figure 3 (b) shows the spectrum of the
+flare emission component, which were obtained by ob-
+taining the difference between the spectrum at each time
+and the pre-flare spectrum. A particularly large blue-
+shifted excess component was observed in the pre-flare
+subtracted spectrum, especially 5 − 18 min, extending
+over −1000 km s−1. Figure 4 (d) presents a color map
+showing the time variation of the blue-shifted excess
+component in the pre-flare subtracted spectrum. The
+blue-shifted excess component was fast over the time
+period when white light and Hα reach their peak.
+While the spectrum before the difference did not do so
+(Figure 3 (a)), the Hα peak of the pre-flare subtracted
+spectrum was slightly red-shifted (∼ +50km s−1; Figure
+3 (b) and 4 (d)). All spectra we present in this paper
+are not corrected for the radial velocity. According to
+Strassmeier (2000), the radial velocity of V1355 Orionis
+varies in the range of about +0 − +70 km s−1. That is,
+the velocity of the red shift is within the range of the
+radial velocity variability. The red shift might be caused
+by the downward chromospheric condensation (Ichimoto
+& Kurokawa 1984) or the post flare-loop (Claes & Kep-
+pens 2019).
+We divided the Hα emission line into flare and blue-
+shifted excess components and then calculated their
+equivalent widths.
+Figure 4 (c) shows their light
+curves.
+The emission lines were separated into flare
+and blue-shifted excess components by fitting only the
+long wavelength side of the lines with the Voigt func-
+tion.
+See Section 4.1 for details about the fitting.
+From the light curve separated into flare and blue-
+shifted excess components, we calculated the energy
+of the flare emitted in Hα (EHα).
+Using the R-band
+magnitude (mR; Cutispoto et al. 1995), R-band Vega
+flux zero point per unit wavelength (fλ = 217.7 ×
+10−11 ergs cm−2 sec−1 ˚A−1; Bessell et al. 1998), and the
+distance between the Earth and V1355 Orionis (d; Gaia
+Collaboration et al. 2016), the equivalent width of the
+flare (EWflare) can be converted to luminosity LHα,
+LHα(t) = 4πd2fλ10−0.4mR × EWflare(t).
+(4)
+Table 1 lists the values of d and mR. By integrating
+LHα with the duration of the Hα flare (∼ 210 min),
+EHα =
+�
+LHα(t) dt = 1.1 × 1034 erg
+(5)
+is obtained.
+4. DISCUSSION
+4.1. Estimation of the prominence parameters
+As discussed in Section 3.2, a clear blue-shifted excess
+component in the Hα emission line was identified during
+the flare. This blue-shifted excess component must re-
+flect a prominence eruption since its velocity extends to
+more than 1000 km s−1. A cool upflow (Tei et al. 2018)
+cannot explain such a rapid blue shift. Therefore, we
+estimated the velocity and mass of the prominence that
+appears as the blue-shifted excess component.
+4.1.1. Velocity
+We estimated the velocity of the prominence using a
+method similar to that used by Maehara et al. (2021).
+As shown in Figure 5 (a) and (c), we first fit the pre-
+flare subtracted spectrum at each time with the Voigt
+function only at wavelengths longer than the line cen-
+ter. Then, we calculated the residuals between the Voigt
+function and the pre-flare subtracted spectrum.
+The
+residual is shown by the blue line in Figure 5 (b) and
+(d). Finally, the residual was fitted with the Gaussian.
+The wavelength of the Gaussian peak was converted to
+the Doppler velocity to evaluate the prominence veloc-
+ity.
+Spectra exist over multiple time periods when the
+residual appeared to have two peaks, as shown in Figure
+5 (b) and (d). Therefore, when fitting the residuals, we
+performed two types of fits: one- and two-component
+Gaussian fits (Figure 5 (b) and (d), respectively). See
+Section 4.2 for the details of and interpretation on why
+the blue-shifted excess component appearing to have two
+peaks.
+The time variation of the velocity of the prominence
+examined in this manner is shown in Figure 6 (d) and
+(h).
+In the case of a one-component fit, the velocity
+of the prominence reached −990 ± 130 km s−1 at the
+peak.
+In the case of a two-component fit, the veloc-
+ity of the prominence reached −1690 ± 100 km s−1 and
+
+A high-velocity prominence eruption on V1355 Orionis
+7
+Figure 4. Light curves and spectra during a superflare on V1355 Orions. (a) Enlarged early part of the light curve of Figure
+2 (a).
+The horizontal and vertical axes indicate the detrended flux and time (unit of minutes) from BJD=2459203.11297,
+respectively. (b) Enlarged Hα light curve of Figure 2 (b) for the same period as in panel (a). The horizontal and vertical axis
+represents the equivalent width of Hα and that the same as in (a), respectively. (c)Hα light curve divided into line center and
+blue-shifted excess components. The pink triangles and blue circles indicate the equivalent widths of the flare and blue-shifted
+excess components, respectively. The equivalent width of the flare component was calculated by integrating the differential
+spectrum from the pre-flare level ±17 ˚A from the Hα line center while blue-shifted excess component was not present. For the
+time period when the emission line was blue-shifted, the equivalent width was calculated by integrating the Voigt function that
+fits the spectrum at longer wavelengths than the peak (e.g., the black dotted line in Figure 5(a)). The equivalent width of the
+blue-shifted excess component was calculated by integrating the residuals between the spectra and the fitted Voigt functions
+(e.g., the blue line in Figure 5(b)). See Section 4.1 for more information about fitting. (d) Time variation of the differential
+spectrum from the pre-flare level at the same time as in (a), (b), and (c). The bottom and top abscissas are the wavelength and
+Doppler velocity from the line center, respectively.
+−760 ± 90 km s−1 at the peak. In the case of both fits,
+the prominence velocity was almost always much faster
+than the escape velocity (−347km s−1) at the surface of
+the K-type star of V1355 Orionis. Therefore, the promi-
+nence that erupted with this flare would certainly have
+flown outward from the star and developed into a CME.
+Moreover, in both cases, the velocity of the blue-
+shifted excess component decelerated more rapidly than
+the gravitational acceleration of the K-type star after
+reaching its peak (see gray lines in Figure 6 (d) and
+(h)). The following are the two possible interpretations.
+The first interpretation is that what we observe in Fig-
+ure 6 (d) and (h) is not the actual deceleration. The
+fast part of the erupted prominence becomes invisible
+rapidly, and the slow part becomes gradually dominant.
+This is observed in the Sun-as-a-star analysis of a fila-
+
+Velocity [km/s]
+- White Light
+Flare
+Hα
+Blue
+-1000
+00
+(a)
+(b)
+(c)
+(d)
+80
+Time from BJD=2459203.11297 [min]
+0.08
+60
+0.06
+40
+0.04
+20
+0.02
+C
+0.00
+65
+8
+6562.8
+Flux
+Wavelength [A]
+E.W. [Al Diff. E.W. [Ai8
+Inoue et al.
+Figure 5. Pre-flare subtracted spectrum and its residual difference from Voigt fitting for the time period when the blue-shifted
+excess component appeared most prominently composed of two components. (a), (c) Pre-flare subtracted spectrum for BJD
+= 2459203.12016776. The black dotted line represents the Voigt function fitted only for the longer wavelength side of the line
+center. Note that the center of the Voigt function is set to the value of the radial velocity of V1355 Orionis calculated from
+the rotational phase. (b), (d) Residual between the observed spectrum and fitting. These spectra correspond to the green line
+minus black dotted line in (a), (c). The black dotted line represents the Gaussian with the residual fitted, shown with (b) one-
+and (d)two-component.
+
+Velocity[km/s]
+-2000
+-1000
+1000
+0.08
+Pre-flare Subtracted
+(a)
+fit (v～ 63 [km/s])
+0.06
+0.04
+D
+0.02
+0.00
+- Residual
+(b)
+0.04
+ fit (Residual)
+enp
+0.02
+Resi
+0.00
+0099
+6620
+6540
+6562.
+Waveiength [A]Velocity [km/s]
+-2000
+-1000
+0.08
+ Pre-flare Subtracted
+(c)
+-• fit (v ～ 63 [km/s])
+0.06
+0.04
+X
+D
+0.02
+0.00
+ Residual
+(d)
+0.04
+fit (Residual)
+enpi
+0.02
+Resic
+0.00
+6520
+6
+00
+6600
+6620
+6562.
+Wavelength [A]A high-velocity prominence eruption on V1355 Orionis
+9
+ment eruption analyzed by Namekata et al. (2022a, see
+the supplementary information therein).
+The second
+interpretation is that the magnetic field applies force
+to the prominence in addition to gravity. Simulations
+conducted by Alvarado-G´omez et al. (2018) have shown
+that the magnetic field of an active star could contribute
+significantly to the slowing of prominences.
+4.1.2. Mass
+We estimated the upper and lower limit prominence
+mass from the equivalent width of the blue-shifted ex-
+cess component. In the method used by Maehara et al.
+(2021), the upper limit of prominence mass is propor-
+tional to the 1.5 power of the area of the region emit-
+ting Hα. Therefore, for a prominence eruption as large
+in scale as this case, the method can significantly over-
+estimate the upper limit of prominence mass because
+the prominence shape would be far from cubic like so-
+lar prominences. So, we improved the method used by
+Maehara et al. (2021).
+As shown in Figure 4 (c), the maximum equivalent
+width of the blue-shifted excess component is ∼ 1 ˚A.
+Converting the equivalent width to luminosity using
+equation (4), the luminosity of the blue-shifted excess
+component Lblue is obtained as
+Lblue ∼ 1 × 1030 erg s−1.
+(6)
+We assume that the NLTE model of the solar promi-
+nence (Heinzel et al. 1994a) can be adapted to the
+present case, and further assume that the optical thick-
+ness of Hα line center τp is 0.1 − 100.
+• τp ∼ 0.1:
+The Hα flux of the prominence per unit time, unit
+area, and unit solid angle FHα is
+FHα ∼ 104 erg s−1 cm−2 sr−1
+(7)
+(see Figure 5 in
+Heinzel et al. 1994a).
+As the
+integral of FHα over the region emitting the Hα
+and the solid angle in the direction toward us is
+Lblue,
+Lblue =
+� �
+FHα dAdΩ = 2πAFHα
+(8)
+where A is the area of the region emitting Hα.
+From equations (6)−(8),
+A ∼ 5 × 1024.5 cm2
+(9)
+is obtained. From equation (7), the emission mea-
+sure n2
+eD of the prominence is
+n2
+eD ∼ 1028 cm−5
+(10)
+(see Figure 15 in
+Heinzel et al. 1994a) where
+D and ne are the geometrical thickness and the
+electron density of the prominence, respectively.
+Though Heinzel et al. (1994a) assumes a range of
+D, FHα and n2
+eD are largely uniquely determined
+for a value of τp without much influence from the
+indefiniteness of D. On the other hand, we need
+to assume values of the electron density ne and the
+hydrogen density nH. The typical electron density
+of solar prominence is
+ne ∼ 1010−11.5 cm−3
+(11)
+(Hirayama 1986). From equations (10) and (11),
+D ∼ 105−8 cm.
+(12)
+Labrosse et al. (2010) showed the ratio between
+the hydrogen density nH and the electron density
+ne of solar prominence is
+ne/nH ∼ 0.2 − 0.9.
+(13)
+From equations (9), (11), (12) and (13), the mass
+of the prominence
+M ∼ mHnHAD
+(14)
+is
+9.5 × 1018 g < M < 1.4 × 1020 g
+(15)
+where mH is the mass of hydrogen atom. The error
+range comes from the assumed range of electron
+density and degree of ionization.
+• τp ∼ 100:
+When the value of τp is ∼ 100,
+FHα ∼ 106 erg s−1 cm−2 sr−1
+(16)
+(see Figure 5 in Heinzel et al. 1994a). Calculated
+as in case τp ∼ 0.1,
+A ∼ 1.6 × 1023 cm2.
+(17)
+Assuming equations (11) and (13) as in case τp ∼
+0.1,
+D ∼ 108−11 cm
+(18)
+9.5 × 1019 g < M < 1.4 × 1021 g.
+(19)
+Combining the ranges of M in equations (15) and (19),
+9.5 × 1018 g < M < 1.4 × 1021 g.
+(20)
+As shown in equations (15) and (19), the upper and
+lower limits of the prominence mass varied by only
+
+10
+Inoue et al.
+Figure 6. Light curves and time variation of the velocity of the blue-shifted excess component. (a),(e) Light curves of white
+light observed with TESS; this is an enlarged version of the light curve shown in Figure 2 (a) only for the time period with
+the blue-shifted excess component visible. (b),(f) Light curves of the equivalent width of Hα observed with KOOLS-IFU on
+the Seimei telescope at the same time as (a), (e). The pink triangles and blue circles indicate the equivalent widths of the flare
+and blue-shifted excess components, respectively. (c),(g) Light curves of the equivalent widths of the components comprising
+the residuals between the Voigt function and pre-flare subtracted spectrum.
+These values were calculated by integrating
+Gaussian functions fitted with residuals. Light blue squares denotes the equivalent width when fitting the residuals with one
+component.(Blue (1)) Medium blue inverted triangles and teal hexagons represent the equivalent widths of the faster and slower
+components, respectively, when the residual is fitted by two components (Blue (2)/(3)). (d),(h) Time variation of the velocity
+of the blue-shifted excess components. Marks are set as in panel (c), (g). The error bars contain two elements: fitting error
+and the variation of the radial velocity of this star. The slopes of the gray lines represent the gravitational acceleration at the
+surface of each of the binary stars. The black dashed lines show the escape velocity of the K-type star of V1355 Orionis.
+
+White Light
+(a)
+(e)
+0.02
+0.02
+xn
+0.00
+0.00
+Flare
+(b)
+(f)
+1.20
+Blue
+1.20
+Ei
+0.00
+0.00
+(g)
+(C)
+1.0
+Blue (1)
+1.0
+Blue (2)
+M
+口
+Blue (3)
+Ei
+0.0
+0.0
+(d)
+(h)
+g (K-type)
+g (K-type)
+-2000
+-2000
+Velocity (1)
+Velocity (2)
+-1600
+-1600
+Velocity (3)
+[km/s]
+g (G-type)
+g (G-type)
+-1200
+-1200
+Velocity
+.800
+-800
+.400
+400
+escape velocity
+5
+10
+15
+20
+25
+0
+5
+10
+15
+20
+25
+0
+Time from BJD=2459203.11297 [min]
+Time from BJD=2459203.11297 [min]A high-velocity prominence eruption on V1355 Orionis
+11
+Figure 7. Schematic diagram that represents the interpretation of the fact that the blue-shifted excess component appears
+to be composed of two components. (a) When two prominences are visible in the Hα emission line. (b) When parts of the
+prominence are visible in the Hα emission line and parts are visible in absorption.
+about an order of magnitude when τp varied signifi-
+cantly. Since we do not know the value of τp, we set
+an extreme value of τp ∼ 100 as the upper limit in this
+paper. The energy of this flare is ∼ 106 times that of a
+typical solar flare (∼ 1030 erg). The energy of a flare is
+proportional to the cube of the spatial scale (Shibata &
+Yokoyama 2002). The typical value of the optical thick-
+ness of the solar prominence is ∼ 1. Simply put, the
+optical thickness of the prominence is proportional to
+the geometric thickness of the prominence. Given the
+spatial scale of the prominence, the upper limit of τp
+can be roughly considered to be ∼ 1 × (106)1/3 = 100.
+On the other hand, the lower limit of τp was limited by
+the hemispheric area of the star. When τp is set to an
+extremely small value, A is extremely larger than the
+hemispherical area of the star (see equations (6) and
+(7)). Such a situation is unrealistic. The value of A in
+equation (9) corresponds to ∼ 101.5πR2. Filling factor
+effect may also affect the prominence mass estimation
+(Kucera et al. 1998). So, a more detailed study of the
+stellar prominence mass calculation is needed in the fu-
+ture.
+4.2. Interpretation of Hα line profile: Two components
+Sometimes the blue-shifted excess component ap-
+peared to have two clear peaks, as shown in Figure 5
+(b) and (d), whereas other times the case is opposite.
+Figure 6 (c) and (g) show the time variation of the equiv-
+alent width of the Gaussian fitted to the residual for the
+one- and two-component fits, respectively.
+As shown
+in Figure 6 (g), the equivalent width of one of the two
+Gaussians was close to zero at the beginning and end
+
+(b)
+Hα Emission
+Prominence
+Flare Ribborn
+Observe
+Star
+Inicident Stellar Radiation
+Hα Absorption(a)
+Flare Ribborn
+Prominence<
+Hα Emission
+Observe
+Star
+Inicident Stellar Radiation12
+Inoue et al.
+Figure 8. Mass, kinetic energy, and velocity of the prominence eruption on V1355 Orionis compared with the statistical data of
+solar and stellar prominence eruptions/CMEs. (a) Comparison between mass of CMEs/prominences and flare energy. The top
+and bottom horizontal axes represent the energy emitted in the GOES wavelength band and bolometric flare energy, respectively.
+Red stars correspond to filament eruptions on the Sun taken from Namekata et al. (2022a). Black crosses correspond to CME
+events on the Sun taken from Yashiro & Gopalswamy (2009). Blue squares indicate stellar mass ejection events on M-dwarfs
+(dMe). Green triangles indicate stellar mass ejection events on young stellar objects (YSO) and close binary systems (CB). Data
+of these stellar events were obtained from Moschou et al. (2019) and Maehara et al. (2021). The orange diamond represents the
+filament eruption event on a young Sun-like star EK Dra reported in Namekata et al. (2022a), wherein a blue-shifted absorption
+component was identified. The pink circle denotes the prominence eruption on V1355 Orionis. The cyan dashed line represents
+the relation: MCME ∝ E2/3
+flare, shown by Takahashi et al. (2016) about the Sun, which is fitted to the solar data points used
+in this study. (b) Velocity of CMEs/prominences as a function of flare energy. Red stars represent the filament eruptions on
+the Sun (obtained from Seki et al. (2019)). The pink circle denotes the velocity of the prominence eruption on V1355 Orionis
+obtained by fitting with one-component Gaussian. The lower and upper limits of the velocity of the prominence eruption on
+V1355 Orionis are the velocity obtained by fitting with two-component Gaussian. The other marks are the same as in (a). The
+scaling law denoted by the cyan dashed line was obtained from Takahashi et al. (2016). Note that this scaling law is an upper
+bound on the speed; thus it has been adjusted to pass through the fastest point in the solar data used here. (c) Comparison
+between kinetic energy of CMEs/prominences and flare energy. The horizontal axis and each mark are the same as those in (a).
+The scaling law denoted by the cyan dashed line was obtained from Namekata et al. (2022a), which is also fitted to the solar
+data points used in this study.
+
+GOES X-ray (1-8 A band) flare energy [erg]
+GOES X-ray (1-8 A band) flare energy [erg]
+1028
+1030
+1032
+1034
+1036
+1026
+1028
+1030
+1032
+1034
+1036
+104
+[(a)
+[(b)
+1022
+16
+1020
+VCME
+103
+Velocity [km/
+1018
+ α Elare
+McME
+S
+Mas
+1016
+1014
+102
+1012
+1030
+1032
+1034
+1036
+1038
+1028
+1030
+1032
+1034
+1036
+1038
+Bolometric flare energy [erg]
+Bolometric flare energy [erg]
+GOES X-ray (l-8 A band) flare energy [erg]
+1028
+1030
+1032
+1034
+1036
+(c)]
+1037
+Solar Filament Eruption
+g
+Solar CME
+05
+dMe (Blue-shift)
+Ekin
+YSO (Blue-shift)
+Kinetic ener
+Young Sun-like Star (Blue-shift Absorption)
+This work (Blue-shift)
+1029
+1025
+1030
+1032
+1034
+1036
+1038
+Bolometric flare energy [erg]A high-velocity prominence eruption on V1355 Orionis
+13
+of the time when the blue-shifted excess component was
+visible.
+Both equivalent widths were above a certain
+value in the intervening. That is, the blue-shifted ex-
+cess component initially appeared to be one component,
+then became two components, and finally appeared to
+be one component again.
+We assume that the prominence visibility for the Sun,
+where the prominence is considered the emission compo-
+nent of Hα and the filament is regarded the absorption
+component of Hα (Parenti 2014), holds true here. Then,
+the fact that the blue-shifted excess component appears
+to be composed of two components can be interpreted
+in two ways:
+(i) Emission + Emission:
+As shown in Figure 7 (a), two prominences are
+present above the limb and each prominence is vis-
+ible as an emission line in Hα. This would be the
+situation when the prominence eruption occurred
+twice, or when the erupted prominence split in two
+while moving.
+(ii) Absorption + Emission:
+As shown in Figure 7 (b), the erupted prominence
+contains both the area above the limb and visible
+as an emission line in Hα, and that inside the limb
+and visible as an absorption line in Hα. When mix-
+ing of those emission and absorption components,
+two peaks seem to exist in the blue-shifted excess
+component. In this case, the width of the emis-
+sion component must be broader than that of the
+absorption component to reproduce the observed
+spectrum. However, the physical interpretation for
+this situation is not well understood.
+Nevertheless, whether the premise of these interpreta-
+tions can be applied to the present case remains unclear.
+That is, unlike the Sun, filaments may be visible as the
+Hα emission components on K-type stars.
+Leitzinger
+et al. (2022) showed that for dM stars, thermal radiation
+from filaments dominates the source function over the
+scattering of the star’s incident radiation so that even
+filaments can be considered emission line components of
+Hα through 1D NLTE modeling and cloud model for-
+mulation. Given that our eruptive event occurred on a
+K-type star, which is cooler than the Sun, a filament
+may not necessarily be visible in the absorption compo-
+nent in Hα, similar to that claimed by Leitzinger et al.
+(2022) for M-type stars. Therefore, modeling and simu-
+lation of prominence/filament eruptions on K-type stars
+are required for a more advanced understanding of the
+two-peaked blue-shifted excess components.
+4.3. Comparison with other events
+We compared the fast prominence eruption observed
+on V1355 Orionis with other events regarding mass, ve-
+locity, and kinetic energy. Figure 8 shows the (a) mass,
+(b) velocity, and (c) kinetic energy of the prominence
+eruptions and CMEs as a function of flare energy emit-
+ted in the GOES wavelength band (1-8 ˚A) and bolomet-
+ric flare energy. We used the equation (1) of Moschou
+et al. (2019) when converting the energy emitted in Hα
+to GOES X-ray flare energy: LX = 16LHα. We also as-
+sumed the relationship: Lbol = 100LX , which is shown
+to hold on solar flares by Emslie et al. (2012). On the
+other hand, Osten & Wolk (2015) shows Lbol = LX/0.06
+for active stars. Therefore, the bolometric energy of the
+stellar flares in Figure 8 may be a bit smaller. Note that
+for stars, only examples estimated from blue-shifted ex-
+cess components of chromospheric lines are plotted in
+Figure 8.
+The lower and upper limits of the velocity
+of the prominence eruption on V1355 Orionis in Fig-
+ure 8 (b) are the velocity obtained by fitting with two-
+component.
+The pink circle denotes the velocity ob-
+tained by fitting with one-component.
+Figure 8 (a) indicates that the erupted prominence on
+V1355 Orionis has roughly the mass that expected from
+the scaling law of solar CMEs. This suggests that the
+prominence eruption on V1355 Orionis was caused by
+the same physical mechanism as the solar prominence
+eruptions/CMEs (e.g., Kotani et al. 2022). Figure 8 (a)
+also shows that it is the largest prominence eruption
+observed by blue shift. These facts may provide impor-
+tant clues as to how large an event can be caused by the
+physical mechanism of solar prominence eruptions. Our
+observations may provide an opportunity to understand
+extreme eruptive events on stars.
+Figure 8 (b) shows that the prominence velocity on
+V1355 Orionis is indeed fast, but overwhelmingly below
+the theoretical upper limit of the velocity (Takahashi
+et al. 2016) estimated from the flare energy. Therefore,
+the fast prominence eruption is physically possible to
+occur in association with a 1035 erg class flare.
+Figure 8 (c) indicates that kinetic energy corresponds
+roughly to the value predicted from the scaling law of
+the solar CMEs. In our calculation, the kinetic energy
+of the prominence is 4.5×1033 erg < K < 1.0×1037 erg.
+As discussed in Moschou et al. (2019), the kinetic en-
+ergy of the prominence on other stars is smaller than
+that expected from the scaling law of solar CMEs. The
+possible reason for the discrepancy is that prominence
+eruptions generally have a lower velocity than CMEs in
+case of the Sun (Maehara et al. 2021; Namekata et al.
+2022a), although it is also proposed that the suppression
+by the overlying large-scale magnetic field can contribute
+to small kinetic energies (Alvarado-G´omez et al. 2018).
+
+14
+Inoue et al.
+For the blue-shift events except on V1355 Orionis, the
+kinetic energy tends to be smaller than the scaling law.
+However, we are not sure if the prominence of V1355
+Orionis is below the scaling law due to the large indefi-
+niteness of the kinetic energy.
+Given these considerations, the prominence eruption
+on V1355 Orionis and solar prominence eruptions may
+have a common physical mechanism. However, the large
+uncertainties in the mass estimate makes it difficult to
+compare them with the solar CME scaling law. For the
+mass estimation, we made various assumptions, as dis-
+cussed in Section 4.1.2, that may be incorrect. A more
+accurate derivation of the prominence mass requires a
+simulation as performed in Leitzinger et al. (2022), as
+well as concerning the two components in Section 4.2.
+5. SUMMARY AND CONCLUSION
+We simultaneously performed spectroscopic observa-
+tions in this study using the Seimei telescope and pho-
+tometric observations using TESS on the RS CVn-type
+star V1355 Orionis. We captured a superflare that re-
+leases 7.0×1035erg and has the following characteristics:
+1. For the first 30 min after the flare started, a pro-
+nounced blue shift in the Hα emission line was
+observed confirming that the prominence eruption
+occurred in association with the flare.
+2. The velocity of the prominence eruption calcu-
+lated from the blue-shift was up to 990 km s−1
+(one-component fitting) and 1690 km s−1 (two-
+component fitting), that is, well above the escape
+velocity of 347 km s−1.
+3. There seems two blue-shifted excess components
+with multiple possible interpretations.
+4. The mass of the prominence eruption is also one
+of the largest ever observed (9.5 × 1018 g < M <
+1.4 × 1021 g), corresponding to the value expected
+from the flare energy-mass scaling law that holds
+for solar CMEs.
+However, the mass estimates
+make many uncertain assumptions and are highly
+indeterminate.
+In a very rare case, a prominence eruption at the ve-
+locity that greatly exceeds the escape velocity of the
+star was captured continuously at a high temporal res-
+olution of 1 min simultaneously with a white light flare.
+The massive and fast prominence eruption detected in
+this study provide an important indicator of how large
+an eruption the physical mechanism of solar prominence
+eruptions can cause at most. Therefore, this will need
+to be investigated in the future with a larger sample of
+prominence eruptions in the energy range of > 1035 erg.
+As also discussed in Leitzinger et al. (2022), sim-
+ply applying the empirical relation of solar promi-
+nence/filament visibility to star can involve many ambi-
+guities. Further, modeling and simulation of prominence
+visibility on K-type stars are necessary to accurately in-
+terpret our data, especially the two-peaked blue-shifted
+excess component. This would also contribute to a more
+accurate derivation of the prominence mass.
+The spectroscopic data used in this study were obtained
+through the program 20B-N-CN03 with the 3.8m Seimei
+telescope, which is located at Okayama Observatory of
+Kyoto University. TESS data were obtained from the
+MAST data archive at the Space Telescope Science Insti-
+tute (STScI). All the TESS data used in this paper can
+be found in MAST: 10.17909/ffwb-dg98 Funding for the
+TESS mission is provided by the NASA Explorer Pro-
+gram. We thank T. Enoto (Kyoto University/RIKEN),
+H. Uchida, and T. Tsuru (Kyoto University) for their
+comments and discussions.
+We acknowledge the In-
+ternational Space Science Institute and the supported
+International Team 510: Solar Extreme Events: Set-
+ting Up a Paradigm (https://www.issibern.ch/teams/
+solextremevent/). This research is supported by JSPS
+KAKENHI grant numbers 20K04032, 20H05643 (H.M.)
+21J00106 (Y.N.), 21J00316 (K.N.) and 21H01131 (H.M.,
+D.N., K.S.). Y.N. was also supported by NASA ADAP
+award program number 80NSSC21K0632 (PI: Adam
+Kowalski).
+Facilities:
+Seimei (Kurita et al. 2020) , TESS
+(Ricker et al. 2015)
+Software:
+astropy (Astropy Collaboration et al.
+2013, 2018), kools ifu red (Matsubayashi et al. 2019),
+IRAF (Tody 1986), PyRAF (Science Software Branch at
+STScI 2012)
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+Floods Relevancy and Identification of Location from
+Twitter Posts using NLP Techniques
+Muhammad Suleman1,†, Muhammad Asif1,†, Tayyab Zamir2,†, Ayaz Mehmood1,†,
+Jebran Khan3, Nasir Ahmad1 and Kashif Ahmad4
+1DCSE, University of Engineering and Technology, Peshawar, Pakistan
+2Abasyn University Islamabad Campus, Pakistan
+3Department of AI, AJOU University, South Korea
+4Department of Computer Science, Munster Technological University, Cork, Ireland
+Abstract
+This paper presents our solutions for the MediaEval 2022 task on DisasterMM. The task is composed of two
+subtasks, namely (i) Relevance Classification of Twitter Posts (RCTP), and (ii) Location Extraction from
+Twitter Texts (LETT). The RCTP subtask aims at differentiating flood-related and non-relevant social posts
+while LETT is a Named Entity Recognition (NER) task and aims at the extraction of location information
+from the text. For RCTP, we proposed four different solutions based on BERT, RoBERTa, Distil BERT, and
+ALBERT obtaining an F1-score of 0.7934, 0.7970, 0.7613, and 0.7924, respectively. For LETT, we used three
+models namely BERT, RoBERTa, and Distil BERTA obtaining an F1-score of 0.6256, 0.6744, and 0.6723,
+respectively.
+1. Introduction
+Natural disasters represent hazardous events that are generally caused by geophysical, hydrological,
+climatological, and meteorological elements. These hazardous events may have an adverse impact
+on human lives and infrastructure. Floods are one such event and it frequently occurs in different
+parts of the world. Similar to other natural disasters, floods may have a significant impact on
+public health and infrastructure. For instance, it has been noticed on numerous occasions that
+roads and communication infrastructure are badly damaged during floods [1].
+A rapid and effective response to disasters may help in mitigating their adverse impact. Access
+to relevant and timely information is critical for an effective response. The literature demonstrates
+several situations where access to relevant information may be possible due to several reasons,
+such as the unavailability of reporters in the area and damage to communication [2]. Recently
+social media and crowdsourcing have been explored as a source of communication, information
+collection, and dissemination in emergency situations. To this aim, several interesting solutions
+have been proposed to collect, analyze, and extract meaningful insights from social media content
+[2]. However, social media content also comes with several limitations. For instance, social media
+content is generally noisy, thus, making access to relevant information very challenging. Similarly,
+MediaEval’22: Multimedia Evaluation Workshop, January 13–15, 2023, Bergen, Norway and Online
+*Corresponding author.
+†These authors contributed equally.
+� [email protected] (K. Ahmad)
+© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
+CEUR
+Workshop
+Proceedings
+http://ceur-ws.org
+ISSN 1613-0073
+CEUR Workshop Proceedings (CEUR-WS.org)
+arXiv:2301.00321v1  [cs.CL]  1 Jan 2023
+
+geolocation information, which is critical for the relevance of the content, is not necessarily
+available for all the relevant posts.
+Considering the importance and applications of social media content in disaster analytics floods
+detection in social media content has been also included in the MediaEval benchmark competition
+as a shared task for several years. This paper presents a solution for the MMDisaster task presented
+in MediaEval 2022 [3]. The challenge aims to solve two key challenges to disaster analytics in
+social media. The first subtask aims at reducing social media noise by automatically filtering
+social media content to obtain relevant content. The second subtask aims at extracting location
+information from social media text, allowing automatic positioning of a potential incident due to
+floods. For both subtasks, we proposed several interesting solutions as described in Section 3.
+2. Related Work
+In recent years, the potential of social media has been widely explored in different application
+domains [4, 5]. Some of the key applications where social media content has been already proven
+very effective include public health [6], education [7], and public resource management [8]. Social
+media outlets have also been widely explored for a diversified set of applications in disasters and
+emergency situations [2]. For instance, Hao et al. [9] proposed a multi-modal framework utilizing
+multi-social media imagery and textual information for damage assessment in disaster-hit areas.
+The key factors analyzed in the work include hazard/disaster type, severity, and damage type.
+Wu et al. [10] also utilized social media data and the associated geo-location information for
+generating early warnings and damage assessment analysis after disasters. Ahmad et al. [11], on
+the other hand, used social media imagery for the analysis of road conditions after the floods.
+More specifically, the authors proposed an early and late-fusion framework to identify passable
+roads in flooded regions. Alam et al. [12] explored the potential of social media content in another
+relevant task of assessing flood severity. To this aim, the authors collected a large-scale benchmark
+dataset namely CrisisMD. The dataset provides a large collection of Twitter posts including textual
+and visual content. Hassan et al. [13] explored a slightly different aspect of natural disasters
+by extracting sentiments and emotions from visual content shared in social media outlets. The
+authors detailed how visual sentiment analysis of disaster-related social media visual content can
+be utilized by different stakeholders, such as news agencies, public authorities, and humanitarian
+organizations.
+Despite being proven very effective in different tasks of disaster analytics, social media content
+has several limitations, such as noisy data and the unavailability of geolocation information. In
+this paper, we propose a solution to overcome such challenges.
+3. Approach
+3.1. Relevance Classification of Twitter Posts (RCTP)
+As a first step, we analyzed the available multimedia content. During the analysis, we observed
+that most of the posts missing visual content. Moreover, most of the images were irrelevant. Thus,
+we decided to use textual information only in our solution. Our framework for the RCTP subtask
+is composed of two steps. In the first step, we performed some pre-processing techniques to clean
+the data by removing unnecessary information, such as usernames, URLs, emojis, and stop words.
+
+After pre-processing, several state-of-the-art NLP algorithms including BERT [14], Roberta [15],
+Distil BERT [16], and ALBERT [17] are used for the classification of the text. Since its a binary
+classification task, in all methods, our cost function is based on binary crossentropy. Moreover, we
+used Adam optimizer with a mini batch size of 32 for 20 epochs.
+3.2. Location Extraction from Twitter Texts (LETT)
+LETT subtask is treated as Named Entity Recognition (NER) task. NER involves locating and
+classifying named entities in text into pre-defined categories [18]. In this task, we are interested
+in the identification of words representing the starting and subsequent words of a text sequence
+referring to a location. In LETT, annotations are provided at the word level. Similar to the RCTP
+task, in this task, we rely on multiple state-of-the-art algorithms including BERT, Roberta, Distil
+BERT, and ALBERT. We note that in this task, since annotations are provided at the word level, we
+did not use any pre-processing technique before training our models.
+3.3. Dataset
+For both subtasks, separate datasets are released. The dataset for RCTP subtask contains data
+from a total of 8,000 tweets. The tweets are collected between May 25, 2020, and June 12, 2020,
+using flood-related keywords in the Italian Language, such as ”alluvione”, ”allagamento”, and
+”esondazione”. The dataset is provided in two different sets namely the development set and the
+test set. The development set is composed of 5,337 tweets while the test set contains a total of
+1,315 tweets.
+The dataset for the LETT subtask is composed of around 6,000 tweets collected between March
+25, 2017, and August 1, 2018, using flood-related Italian keywords. The annotations for this subtask
+are available per word in the tweets.
+4. Results and Analysis
+4.1. Runs Description of RCTP Subtask
+Table 1 shows the experimental results of the proposed solutions on the development set. We note
+that during the experiments on the development set, we used 70% samples of the development
+set for training, 20% for testing, and 10% samples for validation. As can be seen in the table, no
+significant differences can be observed in the performance of the models on the clean and un-clean
+datasets. As far as the performance of the individual models is concerned, slightly better results
+are obtained with BERT compared to the other models. Table 2 provides the official results of the
+proposed solutions on the test set. We note that for the experiments on the test set the models
+are trained on the complete development set. In total, 4 different runs are submitted for the task.
+Our first, second, and fourth runs are based on BERT, RoBERTa, and Distil Bert models trained
+on the un-cleaned dataset, respectively. Our third run is based on the BERT model trained on
+the cleaned dataset. The performance of the models trained on the un-cleaned dataset is higher
+than the models trained on the cleaned dataset. This indicates that the pre-processing information
+resulted in the removal of some relevant features and thus has a negative impact on the results.
+
+Table 1
+Experimental results of RCTP task on the development set.
+Model
+F1-Score on the Clean Dataset
+F1-Score on the Un-clean Dataset
+BERT
+0.95
+0.94
+RoBERTa
+0.94
+0.93
+Distil BERT
+0.93
+0.93
+ALBERT
+0.92
+0.92
+Table 2
+Experimental results of the RCTP task on the test set.
+Run
+Precision
+Recall
+F1-Score
+1 (BERT on Un-clean Dataset)
+0.6949
+0.9251
+0.7934
+2 (RoBERTa on Un-lean Dataset)
+0.6947
+0.9347
+0.7970
+3 (BERT on Clean Dataset)
+0.6486
+0.9213
+0.7613
+4 (Distil BERT on Un-clean Dataset)
+0.6940
+0.9232
+0.7924
+Table 3
+Experimental results of the LETT task on the development set.
+Model
+F1-Score
+BERT
+0.7752
+RoBERTa
+0.8014
+Distil BERT
+0.7658
+ALBERT
+0.6827
+Table 4
+Experimental results of LETT task on the test set.
+Run
+Exact Results
+Partial Results
+Precision
+Recall
+F1-Score
+Precision
+Recall
+F1-Score
+1 (BERT)
+0.596
+0.522
+0.556
+0.628
+0.622
+0.625
+2 (RoBERTa)
+0.540
+0.676
+0.600
+0.577
+0.810
+0.674
+3 (Distil BERT)
+0.563
+0.604
+0.583
+0.610
+0.760
+0.677
+4.2. Runs Description of LETT Subtask
+Table 3 provides the experimental results of the proposed solutions on the development set for
+the LETT subtask. Similar to RCTP, we used 70% samples of the development set for training,
+20% for testing, and 10% samples for validation. A significant variation can be observed in the
+results of the models on the development set. Overall, better results are obtained for RoBerta
+with a significant improvement of 3% over the second-highest results obtained with the BERT
+model. Table 4 provides the official results for the LETT subtask. In the experiments on the test
+set, the models are trained on the complete development set. We note that in the partial results the
+omitted samples are counted as false while in the partial results the omitted samples are completely
+ignored without any penalty. As can be seen in the table, overall, better results are obtained with
+Roberta and Distil BERT compared to the original implementation of the BERT model.
+
+5. Conclusions
+In this paper, we presented our solutions for the DisasterMM challenge posted in MediaEval
+2022. For both subtasks, multiple state-of-the-art NLP algorithms are employed. In the current
+implementation, all the models are used individually, however, we believe these models can
+complement each other if jointly utilized in a merit-based fusion method. In the future, we aim to
+employ different merit-based fusion methods to jointly utilize the capabilities of the individual
+models in both subtasks.
+References
+[1] D. T. Nguyen, F. Ofli, M. Imran, P. Mitra, Damage assessment from social media imagery data
+during disasters, in: Proceedings of international conference on advances in social networks
+analysis and mining 2017, 2017, pp. 569–576.
+[2] N. Said, K. Ahmad, M. Riegler, K. Pogorelov, L. Hassan, N. Ahmad, N. Conci, Natural disasters
+detection in social media and satellite imagery: a survey, Multimedia Tools and Applications
+78 (2019) 31267–31302.
+[3] S. Andreadis, A. Bozas, I. Gialampoukidis, A. Moumtzidou, R. Fiorin, F. Lombardo,
+T. Mavropoulos, D. Norbiato, S. Vrochidis, M. Ferri, I. Kompatsiaris, DisasterMM: Multimedia
+Analysis of Disaster-Related Social Media Data Task at MediaEval 2022, in: Proceedings of
+the MediaEval 2022 Workshop, Bergen, Norway and Online, 2023.
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+[5] I. Alsmadi, K. Ahmad, M. Nazzal, F. Alam, A. Al-Fuqaha, A. Khreishah, A. Algosaibi, Adver-
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+sentiment analysis from disaster images in social media, Sensors 22 (2022) 3628.
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+Transactions on Knowledge and Data Engineering 34 (2020) 50–70.
+

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+page_content=' Munster Technological University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Cork,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Ireland  Abstract  This paper presents our solutions for the MediaEval 2022 task on DisasterMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The task is composed of two  subtasks, namely (i) Relevance Classification of Twitter Posts (RCTP), and (ii) Location Extraction from  Twitter Texts (LETT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The RCTP subtask aims at differentiating flood-related and non-relevant social posts  while LETT is a Named Entity Recognition (NER) task and aims at the extraction of location information  from the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Introduction  Natural disasters represent hazardous events that are generally caused by geophysical, hydrological,  climatological, and meteorological elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Floods are one such event and it frequently occurs in different  parts of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Similar to other natural disasters, floods may have a significant impact on  public health and infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' For instance, it has been noticed on numerous occasions that  roads and communication infrastructure are badly damaged during floods [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  A rapid and effective response to disasters may help in mitigating their adverse impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Access  to relevant and timely information is critical for an effective response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The literature demonstrates  several situations where access to relevant information may be possible due to several reasons,  such as the unavailability of reporters in the area and damage to communication [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Recently  social media and crowdsourcing have been explored as a source of communication, information  collection, and dissemination in emergency situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' To this aim, several interesting solutions  have been proposed to collect, analyze, and extract meaningful insights from social media content  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' However, social media content also comes with several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' For instance, social media  content is generally noisy, thus, making access to relevant information very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Similarly,  MediaEval’22: Multimedia Evaluation Workshop, January 13–15, 2023, Bergen, Norway and Online  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  †These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  � kashif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Use permitted under Creative Commons License Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content='org  ISSN 1613-0073  CEUR Workshop Proceedings (CEUR-WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='org)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='00321v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='CL]  1 Jan 2023  geolocation information, which is critical for the relevance of the content, is not necessarily  available for all the relevant posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Considering the importance and applications of social media content in disaster analytics floods  detection in social media content has been also included in the MediaEval benchmark competition  as a shared task for several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' This paper presents a solution for the MMDisaster task presented  in MediaEval 2022 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The challenge aims to solve two key challenges to disaster analytics in  social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The first subtask aims at reducing social media noise by automatically filtering  social media content to obtain relevant content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The second subtask aims at extracting location  information from social media text, allowing automatic positioning of a potential incident due to  floods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' For both subtasks, we proposed several interesting solutions as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Related Work  In recent years, the potential of social media has been widely explored in different application  domains [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Some of the key applications where social media content has been already proven  very effective include public health [6], education [7], and public resource management [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Social  media outlets have also been widely explored for a diversified set of applications in disasters and  emergency situations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' For instance, Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' [9] proposed a multi-modal framework utilizing  multi-social media imagery and textual information for damage assessment in disaster-hit areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  The key factors analyzed in the work include hazard/disaster type, severity, and damage type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' [10] also utilized social media data and the associated geo-location information for  generating early warnings and damage assessment analysis after disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Ahmad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' [11], on  the other hand, used social media imagery for the analysis of road conditions after the floods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  More specifically, the authors proposed an early and late-fusion framework to identify passable  roads in flooded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' [12] explored the potential of social media content in another  relevant task of assessing flood severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' To this aim, the authors collected a large-scale benchmark  dataset namely CrisisMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The dataset provides a large collection of Twitter posts including textual  and visual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Hassan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' [13] explored a slightly different aspect of natural disasters  by extracting sentiments and emotions from visual content shared in social media outlets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The  authors detailed how visual sentiment analysis of disaster-related social media visual content can  be utilized by different stakeholders, such as news agencies, public authorities, and humanitarian  organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Despite being proven very effective in different tasks of disaster analytics, social media content  has several limitations, such as noisy data and the unavailability of geolocation information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' In  this paper, we propose a solution to overcome such challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Approach  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Relevance Classification of Twitter Posts (RCTP)  As a first step, we analyzed the available multimedia content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' During the analysis, we observed  that most of the posts missing visual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Moreover, most of the images were irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Thus,  we decided to use textual information only in our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Our framework for the RCTP subtask  is composed of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' In the first step, we performed some pre-processing techniques to clean  the data by removing unnecessary information, such as usernames, URLs, emojis, and stop words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  After pre-processing, several state-of-the-art NLP algorithms including BERT [14], Roberta [15],  Distil BERT [16], and ALBERT [17] are used for the classification of the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Since its a binary  classification task, in all methods, our cost function is based on binary crossentropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Moreover, we  used Adam optimizer with a mini batch size of 32 for 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Location Extraction from Twitter Texts (LETT)  LETT subtask is treated as Named Entity Recognition (NER) task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' NER involves locating and  classifying named entities in text into pre-defined categories [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' In this task, we are interested  in the identification of words representing the starting and subsequent words of a text sequence  referring to a location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' In LETT, annotations are provided at the word level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Similar to the RCTP  task, in this task, we rely on multiple state-of-the-art algorithms including BERT, Roberta, Distil  BERT, and ALBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' We note that in this task, since annotations are provided at the word level, we  did not use any pre-processing technique before training our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Dataset  For both subtasks, separate datasets are released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The dataset for RCTP subtask contains data  from a total of 8,000 tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The tweets are collected between May 25, 2020, and June 12, 2020,  using flood-related keywords in the Italian Language, such as ”alluvione”, ”allagamento”, and  ”esondazione”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The dataset is provided in two different sets namely the development set and the  test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The development set is composed of 5,337 tweets while the test set contains a total of  1,315 tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  The dataset for the LETT subtask is composed of around 6,000 tweets collected between March  25, 2017, and August 1, 2018, using flood-related Italian keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The annotations for this subtask  are available per word in the tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Results and Analysis  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Runs Description of RCTP Subtask  Table 1 shows the experimental results of the proposed solutions on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' We note  that during the experiments on the development set, we used 70% samples of the development  set for training, 20% for testing, and 10% samples for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' As can be seen in the table, no  significant differences can be observed in the performance of the models on the clean and un-clean  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' As far as the performance of the individual models is concerned, slightly better results  are obtained with BERT compared to the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Table 2 provides the official results of the  proposed solutions on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' We note that for the experiments on the test set the models  are trained on the complete development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' In total, 4 different runs are submitted for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Our first, second, and fourth runs are based on BERT, RoBERTa, and Distil Bert models trained  on the un-cleaned dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Our third run is based on the BERT model trained on  the cleaned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' The performance of the models trained on the un-cleaned dataset is higher  than the models trained on the cleaned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' This indicates that the pre-processing information  resulted in the removal of some relevant features and thus has a negative impact on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Table 1  Experimental results of RCTP task on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Model  F1-Score on the Clean Dataset  F1-Score on the Un-clean Dataset  BERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='94  RoBERTa  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='93  Distil BERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='93  ALBERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='92  Table 2  Experimental results of the RCTP task on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Run  Precision  Recall  F1-Score  1 (BERT on Un-clean Dataset)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='6949  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='9251  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='7934  2 (RoBERTa on Un-lean Dataset)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='6947  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='9347  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='7970  3 (BERT on Clean Dataset)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='6486  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='9213  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='7613  4 (Distil BERT on Un-clean Dataset)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='6940  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='9232  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='7924  Table 3  Experimental results of the LETT task on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Model  F1-Score  BERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='7752  RoBERTa  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='8014  Distil BERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='7658  ALBERT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='6827  Table 4  Experimental results of LETT task on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  Run  Exact Results  Partial Results  Precision  Recall  F1-Score  Precision  Recall  F1-Score  1 (BERT)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='596  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='522  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='556  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='628  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content='625  2 (RoBERTa)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content='674  3 (Distil BERT)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='563  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content='677  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Runs Description of LETT Subtask  Table 3 provides the experimental results of the proposed solutions on the development set for  the LETT subtask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Similar to RCTP, we used 70% samples of the development set for training,  20% for testing, and 10% samples for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' A significant variation can be observed in the  results of the models on the development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Overall, better results are obtained for RoBerta  with a significant improvement of 3% over the second-highest results obtained with the BERT  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Table 4 provides the official results for the LETT subtask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' In the experiments on the test  set, the models are trained on the complete development set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' We note that in the partial results the  omitted samples are counted as false while in the partial results the omitted samples are completely  ignored without any penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' As can be seen in the table, overall, better results are obtained with  Roberta and Distil BERT compared to the original implementation of the BERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Conclusions  In this paper, we presented our solutions for the DisasterMM challenge posted in MediaEval  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' For both subtasks, multiple state-of-the-art NLP algorithms are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' In the current  implementation, all the models are used individually, however, we believe these models can  complement each other if jointly utilized in a merit-based fusion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' In the future, we aim to  employ different merit-based fusion methods to jointly utilize the capabilities of the individual  models in both subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content='  References  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Norbiato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Pogorelov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Riegler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Conci, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Khreishah, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Antzoulatos, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Pantelidis,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
+page_content=' Ioannidis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Conci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Ofli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+page_content=' Ahmad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQfefg9/content/2301.00321v1.pdf'}
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+AN IMPARTIAL TRANSFORMER FOR STORY VISUALIZATION
+Nikolaos Tsakas
+Maria Lymperaiou
+Giorgos Filandrianos
+Giorgos Stamou
+National Technical University of Athens
+ABSTRACT
+Story Visualization is an advanced task of computed vision
+that targets sequential image synthesis, where the generated
+samples need to be realistic, faithful to their conditioning
+and sequentially consistent. Our work proposes a novel ar-
+chitectural and training approach: the Impartial Transformer
+achieves both text-relevant plausible scenes and sequential
+consistency utilizing as few trainable parameters as possi-
+ble. This enhancement is even able to handle synthesis of
+’hard’ samples with occluded objects, achieving improved
+evaluation metrics comparing to past approaches.
+Index Terms— Story Visualization, GANs, Transformers
+1. INTRODUCTION
+The emergence of GANs [1] has inspired several advance-
+ments in image synthesis, one of the most prominent being
+conditional image synthesis with the usage of cGANs [2].
+Text-conditioned image generation has been a popular vari-
+ant of the conditional case, displaying a long line of archi-
+tectural exploration. Those topics stimulated the novel task
+of Story Visualization (SV), where a visual story needs to be
+generated conditioned on text or other semantic information.
+The images need not only to correspond to their conditioning,
+but also to remain consistent within the sequence, which re-
+quires a global understanding of the story context. The basic
+idea involves a GAN-based variant with one generator G and
+two discriminators. The first discriminator (image discrimi-
+nator Dim) focuses on text-image relevance, while the other
+one (story discriminator Dst) ensures the overall sequential
+coherence. The same task can be viewed as a sequence trans-
+duction problem, a task widely explored with the usage of
+recurrent neural networks (RNNs) and Transformers [3].
+So far, SV has only received a few improvements, while
+it faces scarcity of viable datasets and evaluation methods. To
+this end, we propose a refined transformer-based approach,
+where a simple and lightweight adjustment called Impartial
+transformer is enough to resolve problems present in our pre-
+decessors. A transformer encoder jointly trained from G and
+Dim is employed to create an input representation, yielding
+a resource-friendly scenario comparing to using separate en-
+coders for each generative component or adding a plethora of
+modules [4, 5] to achieve advanced results
+2. RELATED WORK
+Generative Adversarial Networks (GANs) [1] are able to
+synthesize high-quality images by initially receiving random
+noise z ∼ pz in the input of G and are trained to gradually
+improve the synthesized sample from receiving feedback re-
+garding sample quality from D. Conditional GANs (cGANS)
+also receive a conditioning vector y among with z to guide
+synthesis towards certain areas of the target distribution. Ear-
+lier works in conditional synthesis where y is in textual form
+attempt to fully synthesize the final image in one step, re-
+sulting in samples lacking in fidelity [6]. The first significant
+improvements emerged with the introduction of StackGAN
+[7] and its variants [8] which gradually upsample images up
+to the final resolution. Further implementations target detail
+refinement [9, 10] and improvements of text-image relevance
+[11]. Proceeding to the sequential case, StoryGAN [12] intro-
+duced the SV task utilizing RNNs for conditional encoding,
+as well as the two-discriminator GAN architecture that later
+variants follow [13, 14]. Only recently transformer-based ap-
+proaches for conditional encoding emerged [4, 5] indicating
+a new direction of research obeying to recent trends [3].
+3. METHOD
+We propose an updated framework for the SV task based on
+the emergence of transformer-based techniques for sequence
+processing.
+Primarily, we recommend the use of a trans-
+former encoder [3] as a replacement for the RNN structure
+of StoryGAN [12], focusing on its optimal training regime.
+Fig. 1: The generator G network (T = 4 frames)
+arXiv:2301.03563v1  [cs.CV]  9 Jan 2023
+
+FullyConnected
+Residual
+Upsampling
+Convolution 3x3
+Attention
+(optional)
+Noise ~ Z
+Embedding
+CA
+Output
+Input
+Transformer
+Encoder
+Sentence
+Upsampling
+Image3.1. Generator
+The input to the generator G is a sequence of symbols st,
+embedded by an encoder into vector representations φt, t ∈
+[1, T] where T corresponds to the length of all stories. Fig. 1
+depicts the basic G architecture.
+We recommend using a conditioning augmentation (CA)
+module, similar to [7]: Instead of conditioning the GAN on
+an embedding of the input φt, a random vector ˆc is sampled
+from a Gaussian distribution N(µ(φt, Σ(φt))) with the mean
+µ(φt) and the diagonal covariance matrix Σ(φt) being func-
+tions of the input embeddings. The vector ˆc serves as the
+conditioning variable. CA promotes continuity in the data
+manifold, and can be also used to map the dimension of φt to
+its appropriate size. Training the parameters of this stochas-
+tic process becomes possible using the reparametrization trick
+[15], where a sample from a Gaussian distribution with arbi-
+trary mean µ and covariance matrix σ can be produced as:
+ˆc = µ + z ∗ σ, where z ∼ N(0, 1). In addition, to ensure the
+smoothness of the manifold, the KL divergence between the
+learned Gaussian distribution and the standard one is added
+to the loss function of G as a regularization term, therefore
+avoiding overfitting caused by collapsing to a single point or
+by a distribution that deviates from the standard Gaussian [7]:
+LossKL = DKL(N(µ(ϕt), Σ(ϕt))∥N(0, I))
+The Transformer inputs ˆct are first added to positional en-
+codings to properly influence transduction, and then context-
+aware conditioning vectors ct are produced from the position
+encoded inputs. The context-informed vectors ct are concate-
+nated with Gaussian noise zt ∼ pz, where pz is the random
+input prior z ∼ N(0, 1). This combined input is fed through a
+fully connected (FC) layer, mapping each instance to dimen-
+sion C × H × W, where H, W are the height and width of
+the initial image channels to be upsampled, and C their chan-
+nel number. This output mapping is rearranged in a tensor
+It ∈ RC×H×W and fed through a set of residual upsampling
+blocks, similar to [16]. The purpose of a residual block [17]
+is to learn a mapping F(x) = H(x) − x where H(x) is the
+actual desired mapping in the underlying distribution. The fi-
+nal output is produced utilizing a skip connection such that
+ˆH(x) = F(x) + x. In each upsampling block, the input im-
+age features It are normalized via Batch Normalization [18]
+and passed through a ReLU activation. Then, both spatial di-
+mensions are doubled via nearest-neighbor upsampling, and
+a convolutional filter is applied to transform image features,
+while halving the channel dimension to mitigate computa-
+tional complexity as the image planes get larger. The tensor
+is again normalized and passed through a ReLU activation as
+well as a final convolutional filter. In order to match the spa-
+tial input and output dimensions we perform a minimal trans-
+form on the skip connection, using nearest-neighbor upsam-
+pling and passing through a learned 1 × 1 convolutional filter.
+After feature upsampling to the desired dimension H × W, a
+Fig. 2: Image discriminator Dim (T = 4 frames)
+final 3 × 3 convolution layer is used to produce a 3-channel
+image, followed by a tanh activation to remap pixel values
+into [−1, 1]. We also use Spectral Normalization to further
+stabilize the training process. The entire image sequence can
+be generated in parallel, greatly improving training efficiency.
+3.2. Image Discriminator
+The image discriminator Dim (Fig. 2) is tasked to discern
+between real and generated images individually. To that end,
+Dim utilizes the input features φt of each individual sentence
+corresponding to a story frame, the context, and the image
+It itself to be evaluated. The context is important for Dim,
+because each frame in a story depends on the rest to form
+many of its details. Each image to be evaluated is passed
+through a series of residual downsampling blocks. Image fea-
+tures from each layer are first passed through a Leaky ReLU,
+then from a spectrally normalized convolutional layer, remap-
+ping the C × H × W tensor to double the channels. Af-
+ter another Leaky ReLU, a spectrally normalized strided con-
+volution layer downsamples the image features. We prefer
+this option over a pooling layer due to the inferences made
+by Radford et.
+al in [19].
+All images are evaluated in a
+batch to take advantage of the Transformer’s parallel process-
+ing. Dropout in all Dim residual blocks is proven beneficial,
+to prevent overfitting and overt coupling of individual layer
+units. To produce an output scalar, each vector of dimension
+dmodel given by the encoder is spatially replicated to create a
+dmodel ×H ×W tensor that is then concatenated with the im-
+age features along the channel axis. These features are passed
+through a residual block to jointly learn from image and text
+features. A final FC layer mapping features to a single scalar
+leads to a sigmoid activation function, ultimately producing a
+probability Dim(It) ∈ [0, 1].
+3.3. Story Discriminator
+The story discriminator Dst (Fig. 3) enforces consistency
+and meaningful progression along the image sequence I =
+(I1, ..., IT ) by jointly learning a common feature space for
+text and images. The image features are downsampled using
+similar residual blocks as in Dim. All image features for the
+
+Input
+Embedding
+Transformer
+Spatial replication
+indino
+Spatially
+Sentence
+replicated
+Residual Block
+Fully Conncected
+Repeat
+text features
+dmodel × H × W
+Scalar
+Image
+Image
+(real/fake)
+Downsampling
+features
+C xH xW
+(2C) × (H/2) × (W/2)
+Input
+Residual
+Downsampling
+Attention
+(optional)Fig. 3: Story discriminator Dst (T = 4 frames)
+same story are concatenated into a single storyboard vector.
+On the text side, a FC layer maps all sentence embeddings
+S = (φ1, ..., φT ) to vectors in this shared space, also concate-
+nated into one big text feature vector. The two story-wide vec-
+tors are then multiplied elementwise and the result is passed
+through a FC layer to output a scalar similarity score Dst.
+3.4. Training
+Training requires minimizing Lim, Lst, LG:
+Lim =
+T
+�
+t=1
+(E(it,ϕt)[logDim(it, ϕt, h0; ψI)]+
+E(zt,ϕt)[log(1 − Dim(G(zt, ϕt; θ), ϕt, h0; ψI))]),
+Lst = E(I,S)[logDst(I, S; ψS)]+
+Eϵ,S[log(1 − Dst([G(zt, ϕt; θ)]T
+t=1), S; ψS))],
+LG = E(zt,ϕt)[log(Dim(G(zt, ϕt; θ), ϕt, h0; ψI))]+
+Eϵ,S[log(Dst([G(zt, ϕt; θ)]T
+t=1), S; ψS))] + LossKL
+where zt ∼ pz, and h0 serves as story embedding. The alter-
+native formulation following [1] is employed for G to provide
+sufficient gradients. We also use the matching aware discrim-
+inator criterion as in [20]. One-sided label smoothing is uti-
+lized by setting positive labels to 0.9 instead of 1.0 to avoid
+the pitfalls of regular label smoothing [21].
+4. EXPERIMENTS
+We present results on CLEVR-SV [22], focusing on cases
+where objects may not be clearly separated or even occluded.
+This issue, despite its significance, was not addressed in prior
+work. For all experiments, Adam optimizer [23] is used for
+gradient descent with β1 = 0.5 and β2 = 0.999. After exten-
+sive hyperparameter tuning we present results on the original
+Transformer with dmodel = 512, Nheads = 8, Nlayers = 6.
+4.1. Impartial Transformer Encoder
+We explore the option of utilizing one Impartial transformer
+encoder, whose parameters are updated jointly by G and
+Dim. We hypothesize such an encoder would learn a task-
+conducive representation for embedding sequences by simply
+encoding necessary context without giving an advantage to
+either adversary. We further attempted to train the encoder to
+also receive gradients from the Dst, but found this addition to
+be confusing the encoder, to the point of learning completely
+mismatched representations of the context space.
+4.2. Learning rate schemes
+Motivated by the Two Time-scale Update Rule [24], we at-
+tempt to find an optimal learning rate scheme for the three
+networks while maintaining a 1/1/1 update ratio for more ef-
+ficient training, thus proposing a Three Time-scale Update
+Rule. After 20 epochs, the learning rates are halved based on
+a typical scheduling scheme. We observe that when G learns
+faster than the discriminators, the whole model suffers from
+mode collapse: G easily fools both discriminators early on,
+leading training to a stalemate since the discriminators can-
+not produce any meaningful gradients to guide generation.
+When maintaining a low learning rate for G, increasing the
+Dim learning rate proves to lead G into creating images that
+correspond better to the conditioning. G is faster in learning
+the correct matching for color and shape between image and
+description vector, as well as learning to produce more con-
+crete shape features, at least for large objects. When increas-
+ing the learning rate of Dst, we immediately observe greater
+consistency across images. Lower learning rates also seem
+to affect text-image matching, with G creating images with
+wrong color, shape and size more frequently. We thus argue
+that it is beneficial for the two discriminators to learn about 4
+times as fast as G. Specifically, we find lrG = 0.0001, lrDim
+= 0.0004, lrDst = 0.0004 to be optimal, as higher learning
+rates proved to be too fast for convergence.
+4.3. Warmup Scheduler
+We experiment with decaying the learning rate by halving it
+every 20 epochs. The original Transformer [3] recommends
+a specific learning rate scheduling scheme to be used along
+with the Adam optimizer: The learning rate should first be
+increased linearly for a number of warmup steps and then de-
+creased proportionally to the inverse square root of the num-
+ber of total steps, where one step is considered to be a sin-
+gle batch of data passing through the network. We observe
+that the scheduler fails to train the context encoder, result-
+ing in mostly nonsensical representations. We presume this is
+because the recommended optimizer only takes into account
+dmodel and the number of warmup steps, forcing the learning
+rate to generally remain much higher than what the learning
+rates of the Adam optimizer in regular decay are, preventing
+network from convergence.
+4.4. Results
+Visual results including ablations are presented in Fig 4, while
+comparison over easy and hard examples are presented in Fig.
+5. There is an obvious improvement over StoryGAN [12],
+which fails to generate the proper sequence, and also lacks in
+
+Input
+Embedding
+FC
+Text
+Vector
+Fully Connected
+Sentence
+FC
+Input
+Elementwise
+FC
+Scalar
+product
+Image 
+Vector
+Output
+Image
+Downsampling(a) Left: Ground truth (T=4). Middle: StoryGAN generated frames, low relevance and object quality. Right: Ours, baseline.
+(b) Our results without attention. Left: Separate Transformer Encoder for G, Dim, Dst, low object relevance. Middle: Impartial
+Encoder (G and Dim gradients). Right: Impartial encoder (all G, Dim, Dst gradients), mode collapse.
+Fig. 4: Ablation studies of our framework indicate the power of the Impartial Transformer (G and Dim gradients).
+Fig. 5: (a) 1st row ground truth, (b) 2nd row generated frames (ours-Impartial Transformer), (c) 3rd row generated frames
+(storyGAN) of 3 stories with T=4. From left to right (every 4 images) difficulty of stories increases due to object occlusion.
+fidelity. The second row of Fig. 4 indicates the optimal usage
+of the Impartial transformer. Even though our implementa-
+tion presents satisfactory results when objects are placed in a
+distance from each other (Fig 5, left), in cases when objects
+are adjacent or overlap, there are some sacrifices to be made:
+either semantics -especially shape and material- are not dis-
+tinct enough (Fig 5, middle), or objects are ’swallowed’ by
+their neighbors (Fig 5, right), which results in low quality se-
+mantics. The results of human evaluation experiments over
+preference are presented in Table 1. Results using automated
+metrics are presented in Table 2. Our framework clearly out-
+performs prior efforts [12, 4, 5] according to Clean-FID [25],
+LPIPS [26] and SSIM. We mainly focus on LPIPS metric for
+comparison that reflects human perception, where we achieve
+16% improvement over prior approaches [12, 4, 5].
+Table 1: Human Evaluation preference (averaged results),
+Win% = % times our output stories were preferred over [12],
+Lose% for vice-versa, Tie% when equally preferred.
+Attribute
+Win%
+Loose%
+Tie%
+Visual Quality
+25
+20
+55
+Consistency
+37
+32
+31
+Relevance
+32
+30
+38
+Table 2: Average evaluation metrics.
+Frame FID↓
+Clean-
+FID↓
+LPIPS↓
+SSIM↑
+1st
+32.94 ± 7.85
+111.20
+0.18 ± 0.06
+0.81
+2nd
+37.41 ± 6.67
+110.80
+0.19 ± 0.05
+0.73
+3rd
+47.41 ± 15.83
+106.69
+0.23 ± 0.05
+0.68
+4th
+48.41 ± 3.84
+133.15
+0.25 ± 0.05
+0.62
+All
+41.54 ± 8.55
+115.46
+0.21 ± 0.05
+0.71
+[12]
+41.45 ± 6.25
+123.40
+0.25 ± 0.03
+0.65
+[5]
+41.96 ± 9.66
+124.97
+0.25 ± 0.08
+0.67
+[4]
+41.80 ± 8.81
+122.62
+0.25 ± 0.05
+0.68
+’All’ refers to global results of the Impartial Transformer and is compare
+with the global results of [12], [5], [4]. Results from [5], [4] are obtained by
+re-training on CLEVR-SV.
+5. CONCLUSION
+In this work, we developed a transformer-inspired framework
+for story visualization, aiming to set a new baseline in litera-
+ture by achieving improvements according to perceptual met-
+rics. The usage of the Impartial Transformer demonstrated
+promising directions for the evolution of generative models
+in the same track, as few -if any- current implementations ex-
+ploit a ’forking’ module jointly trained by two adversaries.
+As future work we plan to explore the evaluation part of SV.
+
+6. REFERENCES
+[1] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza,
+Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron
+Courville, and Yoshua Bengio, “Generative adversarial
+nets,” in NeurIPS, 2014.
+[2] Augustus Odena,
+Christopher Olah,
+and Jonathon
+Shlens,
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+classifier gans,” 2017.
+[3] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob
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+Kaiser, and Illia Polosukhin, “Attention is all you need,”
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+[4] Adyasha Maharana, Darryl Hannan, and Mohit Bansal,
+“Improving generation and evaluation of visual stories
+via semantic consistency,” ArXiv, vol. abs/2105.10026,
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+[5] Adyasha Maharana and Mohit Bansal, “Integrating vi-
+suospatial, linguistic, and commonsense structure into
+story visualization,” ArXiv, vol. abs/2110.10834, 2021.
+[6] Scott E. Reed, Zeynep Akata, Xinchen Yan, Lajanugen
+Logeswaran, Bernt Schiele, and Honglak Lee, “Gener-
+ative adversarial text to image synthesis,” CoRR, vol.
+abs/1605.05396, 2016.
+[7] Han Zhang, Tao Xu, and Hongsheng Li,
+“Stackgan:
+Text to photo-realistic image synthesis with stacked gen-
+erative adversarial networks,” in ICCV, 2017.
+[8] Han Zhang, Tao Xu, Hongsheng Li, Shaoting Zhang,
+Xiaogang Wang, Xiaolei Huang, and Dimitris Metaxas,
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+generative adversarial networks,” 2018.
+[9] Tao Xu, Pengchuan Zhang, Qiuyuan Huang, Han
+Zhang, Zhe Gan, Xiaolei Huang, and Xiaodong He, “At-
+tngan: Fine-grained text to image generation with atten-
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+[10] Minfeng Zhu, Pingbo Pan, Wei Chen, and Yi Yang,
+“DM-GAN: dynamic memory generative adversarial
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+CoRR, vol.
+abs/1904.01310, 2019.
+[11] Hongchen Tan, Xiuping Liu, Xin Li, Yi Zhang, and Bao-
+cai Yin, “Semantics-enhanced adversarial nets for text-
+to-image synthesis,” in ICCV, 2019.
+[12] Yitong Li, Zhe Gan, Yelong Shen, Jingjing Liu,
+Yu Cheng, Yuexin Wu, Lawrence Carin, David Edwin
+Carlson, and Jianfeng Gao,
+“Storygan: A sequential
+conditional gan for story visualization,” CVPR, 2019.
+[13] Gangyan Zeng, Zhaohui Li, and Yuan Zhang, “Pororo-
+gan: An improved story visualization model on pororo-
+sv dataset,” CSAI2019, 2019, ACM.
+[14] Chunye Li, Liya Kong, and Zhiping Zhou, “Improved-
+storygan for sequential images visualization,” Journal
+of Visual Communication and Image Representation,
+2020.
+[15] Diederik P Kingma and Max Welling, “Auto-encoding
+variational bayes,” 2014.
+[16] Han Zhang, Ian Goodfellow, Dimitris Metaxas, and Au-
+gustus Odena,
+“Self-attention generative adversarial
+networks,” 2018.
+[17] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian
+Sun, “Deep residual learning for image recognition,” in
+CVPR, 2016.
+[18] Sergey Ioffe and Christian Szegedy, “Batch normaliza-
+tion: Accelerating deep network training by reducing
+internal covariate shift,” 2015.
+[19] Alec Radford, Luke Metz, and Soumith Chintala, “Un-
+supervised representation learning with deep convolu-
+tional generative adversarial networks,” 2016.
+[20] Scott Reed, Zeynep Akata, Xinchen Yan, Lajanugen Lo-
+geswaran, Bernt Schiele, and Honglak Lee, “Generative
+adversarial text to image synthesis,” in ICML, 2016.
+[21] Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe,
+Jonathon Shlens, and Zbigniew Wojna,
+“Rethinking
+the inception architecture for computer vision,” CVPR,
+2016.
+[22] Justin Johnson, Bharath Hariharan, Laurens van der
+Maaten, Li Fei-Fei, C. Lawrence Zitnick, and Ross B.
+Girshick,
+“Clevr:
+A diagnostic dataset for com-
+positional language and elementary visual reasoning,”
+CVPR, 2017.
+[23] Diederik Kingma and Jimmy Ba, “Adam: A method for
+stochastic optimization,” ICLR, 2014.
+[24] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner,
+Bernhard Nessler, and Sepp Hochreiter, “Gans trained
+by a two time-scale update rule converge to a local nash
+equilibrium,” 2017.
+[25] Gaurav Parmar, Richard Zhang, and Jun-Yan Zhu, “On
+aliased resizing and surprising subtleties in gan evalua-
+tion,” in CVPR, 2022.
+[26] Richard Zhang, Phillip Isola, Alexei A Efros, Eli
+Shechtman, and Oliver Wang, “The unreasonable ef-
+fectiveness of deep features as a perceptual metric,” in
+CVPR, 2018.
+

DNE1T4oBgHgl3EQf-AaO/content/tmp_files/load_file.txt

@@ -0,0 +1,254 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf,len=253
+page_content='AN IMPARTIAL TRANSFORMER FOR STORY VISUALIZATION  Nikolaos Tsakas  Maria Lymperaiou  Giorgos Filandrianos  Giorgos Stamou  National Technical University of Athens  ABSTRACT  Story Visualization is an advanced task of computed vision  that targets sequential image synthesis, where the generated  samples need to be realistic, faithful to their conditioning  and sequentially consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Our work proposes a novel ar-  chitectural and training approach: the Impartial Transformer  achieves both text-relevant plausible scenes and sequential  consistency utilizing as few trainable parameters as possi-  ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' This enhancement is even able to handle synthesis of  ’hard’ samples with occluded objects, achieving improved  evaluation metrics comparing to past approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Index Terms— Story Visualization, GANs, Transformers  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' INTRODUCTION  The emergence of GANs [1] has inspired several advance-  ments in image synthesis, one of the most prominent being  conditional image synthesis with the usage of cGANs [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Text-conditioned image generation has been a popular vari-  ant of the conditional case, displaying a long line of archi-  tectural exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Those topics stimulated the novel task  of Story Visualization (SV), where a visual story needs to be  generated conditioned on text or other semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  The images need not only to correspond to their conditioning,  but also to remain consistent within the sequence, which re-  quires a global understanding of the story context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The basic  idea involves a GAN-based variant with one generator G and  two discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The first discriminator (image discrimi-  nator Dim) focuses on text-image relevance, while the other  one (story discriminator Dst) ensures the overall sequential  coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The same task can be viewed as a sequence trans-  duction problem, a task widely explored with the usage of  recurrent neural networks (RNNs) and Transformers [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  So far, SV has only received a few improvements, while  it faces scarcity of viable datasets and evaluation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' To  this end, we propose a refined transformer-based approach,  where a simple and lightweight adjustment called Impartial  transformer is enough to resolve problems present in our pre-  decessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' A transformer encoder jointly trained from G and  Dim is employed to create an input representation, yielding  a resource-friendly scenario comparing to using separate en-  coders for each generative component or adding a plethora of  modules [4, 5] to achieve advanced results  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' RELATED WORK  Generative Adversarial Networks (GANs) [1] are able to  synthesize high-quality images by initially receiving random  noise z ∼ pz in the input of G and are trained to gradually  improve the synthesized sample from receiving feedback re-  garding sample quality from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Conditional GANs (cGANS)  also receive a conditioning vector y among with z to guide  synthesis towards certain areas of the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Ear-  lier works in conditional synthesis where y is in textual form  attempt to fully synthesize the final image in one step, re-  sulting in samples lacking in fidelity [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The first significant  improvements emerged with the introduction of StackGAN  [7] and its variants [8] which gradually upsample images up  to the final resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Further implementations target detail  refinement [9, 10] and improvements of text-image relevance  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Proceeding to the sequential case, StoryGAN [12] intro-  duced the SV task utilizing RNNs for conditional encoding,  as well as the two-discriminator GAN architecture that later  variants follow [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Only recently transformer-based ap-  proaches for conditional encoding emerged [4, 5] indicating  a new direction of research obeying to recent trends [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' METHOD  We propose an updated framework for the SV task based on  the emergence of transformer-based techniques for sequence  processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Primarily, we recommend the use of a trans-  former encoder [3] as a replacement for the RNN structure  of StoryGAN [12], focusing on its optimal training regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 1: The generator G network (T = 4 frames)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='03563v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='CV]  9 Jan 2023  FullyConnected  Residual  Upsampling  Convolution 3x3  Attention  (optional)  Noise ~ Z  Embedding  CA  Output  Input  Transformer  Encoder  Sentence  Upsampling  Image3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Generator  The input to the generator G is a sequence of symbols st,  embedded by an encoder into vector representations φt, t ∈  [1, T] where T corresponds to the length of all stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 1  depicts the basic G architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  We recommend using a conditioning augmentation (CA)  module, similar to [7]: Instead of conditioning the GAN on  an embedding of the input φt, a random vector ˆc is sampled  from a Gaussian distribution N(µ(φt, Σ(φt))) with the mean  µ(φt) and the diagonal covariance matrix Σ(φt) being func-  tions of the input embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The vector ˆc serves as the  conditioning variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' CA promotes continuity in the data  manifold, and can be also used to map the dimension of φt to  its appropriate size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Training the parameters of this stochas-  tic process becomes possible using the reparametrization trick  [15], where a sample from a Gaussian distribution with arbi-  trary mean µ and covariance matrix σ can be produced as:  ˆc = µ + z ∗ σ, where z ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' to ensure the  smoothness of the manifold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' the KL divergence between the  learned Gaussian distribution and the standard one is added  to the loss function of G as a regularization term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' therefore  avoiding overfitting caused by collapsing to a single point or  by a distribution that deviates from the standard Gaussian [7]:  LossKL = DKL(N(µ(ϕt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Σ(ϕt))∥N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' I))  The Transformer inputs ˆct are first added to positional en-  codings to properly influence transduction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' and then context-  aware conditioning vectors ct are produced from the position  encoded inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The context-informed vectors ct are concate-  nated with Gaussian noise zt ∼ pz, where pz is the random  input prior z ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' This combined input is fed through a  fully connected (FC) layer, mapping each instance to dimen-  sion C × H × W, where H, W are the height and width of  the initial image channels to be upsampled, and C their chan-  nel number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' This output mapping is rearranged in a tensor  It ∈ RC×H×W and fed through a set of residual upsampling  blocks, similar to [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The purpose of a residual block [17]  is to learn a mapping F(x) = H(x) − x where H(x) is the  actual desired mapping in the underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The fi-  nal output is produced utilizing a skip connection such that  ˆH(x) = F(x) + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' In each upsampling block, the input im-  age features It are normalized via Batch Normalization [18]  and passed through a ReLU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Then, both spatial di-  mensions are doubled via nearest-neighbor upsampling, and  a convolutional filter is applied to transform image features,  while halving the channel dimension to mitigate computa-  tional complexity as the image planes get larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The tensor  is again normalized and passed through a ReLU activation as  well as a final convolutional filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' In order to match the spa-  tial input and output dimensions we perform a minimal trans-  form on the skip connection, using nearest-neighbor upsam-  pling and passing through a learned 1 × 1 convolutional filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  After feature upsampling to the desired dimension H × W, a  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 2: Image discriminator Dim (T = 4 frames)  final 3 × 3 convolution layer is used to produce a 3-channel  image, followed by a tanh activation to remap pixel values  into [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We also use Spectral Normalization to further  stabilize the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The entire image sequence can  be generated in parallel, greatly improving training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Image Discriminator  The image discriminator Dim (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 2) is tasked to discern  between real and generated images individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' To that end,  Dim utilizes the input features φt of each individual sentence  corresponding to a story frame, the context, and the image  It itself to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The context is important for Dim,  because each frame in a story depends on the rest to form  many of its details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Each image to be evaluated is passed  through a series of residual downsampling blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Image fea-  tures from each layer are first passed through a Leaky ReLU,  then from a spectrally normalized convolutional layer, remap-  ping the C × H × W tensor to double the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Af-  ter another Leaky ReLU, a spectrally normalized strided con-  volution layer downsamples the image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We prefer  this option over a pooling layer due to the inferences made  by Radford et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  al in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  All images are evaluated in a  batch to take advantage of the Transformer’s parallel process-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Dropout in all Dim residual blocks is proven beneficial,  to prevent overfitting and overt coupling of individual layer  units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' To produce an output scalar, each vector of dimension  dmodel given by the encoder is spatially replicated to create a  dmodel ×H ×W tensor that is then concatenated with the im-  age features along the channel axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' These features are passed  through a residual block to jointly learn from image and text  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' A final FC layer mapping features to a single scalar  leads to a sigmoid activation function, ultimately producing a  probability Dim(It) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Story Discriminator  The story discriminator Dst (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 3) enforces consistency  and meaningful progression along the image sequence I =  (I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=', IT ) by jointly learning a common feature space for  text and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The image features are downsampled using  similar residual blocks as in Dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' All image features for the  Input  Embedding  Transformer  Spatial replication  indino  Spatially  Sentence  replicated  Residual Block  Fully Conncected  Repeat  text features  dmodel × H × W  Scalar  Image  Image  (real/fake)  Downsampling  features  C xH xW  (2C) × (H/2) × (W/2)  Input  Residual  Downsampling  Attention  (optional)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 3: Story discriminator Dst (T = 4 frames)  same story are concatenated into a single storyboard vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  On the text side, a FC layer maps all sentence embeddings  S = (φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=', φT ) to vectors in this shared space, also concate-  nated into one big text feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The two story-wide vec-  tors are then multiplied elementwise and the result is passed  through a FC layer to output a scalar similarity score Dst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Training  Training requires minimizing Lim, Lst, LG:  Lim =  T  �  t=1  (E(it,ϕt)[logDim(it, ϕt, h0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' ψI)]+  E(zt,ϕt)[log(1 − Dim(G(zt, ϕt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' θ), ϕt, h0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' ψI))]),  Lst = E(I,S)[logDst(I, S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' ψS)]+  Eϵ,S[log(1 − Dst([G(zt, ϕt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' θ)]T  t=1), S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' ψS))],  LG = E(zt,ϕt)[log(Dim(G(zt, ϕt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' θ), ϕt, h0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' ψI))]+  Eϵ,S[log(Dst([G(zt, ϕt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' θ)]T  t=1), S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' ψS))] + LossKL  where zt ∼ pz, and h0 serves as story embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The alter-  native formulation following [1] is employed for G to provide  sufficient gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We also use the matching aware discrim-  inator criterion as in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' One-sided label smoothing is uti-  lized by setting positive labels to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='9 instead of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='0 to avoid  the pitfalls of regular label smoothing [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' EXPERIMENTS  We present results on CLEVR-SV [22], focusing on cases  where objects may not be clearly separated or even occluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  This issue, despite its significance, was not addressed in prior  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' For all experiments, Adam optimizer [23] is used for  gradient descent with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='5 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' After exten-  sive hyperparameter tuning we present results on the original  Transformer with dmodel = 512, Nheads = 8, Nlayers = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Impartial Transformer Encoder  We explore the option of utilizing one Impartial transformer  encoder, whose parameters are updated jointly by G and  Dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We hypothesize such an encoder would learn a task-  conducive representation for embedding sequences by simply  encoding necessary context without giving an advantage to  either adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We further attempted to train the encoder to  also receive gradients from the Dst, but found this addition to  be confusing the encoder, to the point of learning completely  mismatched representations of the context space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Learning rate schemes  Motivated by the Two Time-scale Update Rule [24], we at-  tempt to find an optimal learning rate scheme for the three  networks while maintaining a 1/1/1 update ratio for more ef-  ficient training, thus proposing a Three Time-scale Update  Rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' After 20 epochs, the learning rates are halved based on  a typical scheduling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We observe that when G learns  faster than the discriminators, the whole model suffers from  mode collapse: G easily fools both discriminators early on,  leading training to a stalemate since the discriminators can-  not produce any meaningful gradients to guide generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  When maintaining a low learning rate for G, increasing the  Dim learning rate proves to lead G into creating images that  correspond better to the conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' G is faster in learning  the correct matching for color and shape between image and  description vector, as well as learning to produce more con-  crete shape features, at least for large objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' When increas-  ing the learning rate of Dst, we immediately observe greater  consistency across images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Lower learning rates also seem  to affect text-image matching, with G creating images with  wrong color, shape and size more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We thus argue  that it is beneficial for the two discriminators to learn about 4  times as fast as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Specifically, we find lrG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='0001, lrDim  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='0004, lrDst = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='0004 to be optimal, as higher learning  rates proved to be too fast for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Warmup Scheduler  We experiment with decaying the learning rate by halving it  every 20 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The original Transformer [3] recommends  a specific learning rate scheduling scheme to be used along  with the Adam optimizer: The learning rate should first be  increased linearly for a number of warmup steps and then de-  creased proportionally to the inverse square root of the num-  ber of total steps, where one step is considered to be a sin-  gle batch of data passing through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We observe  that the scheduler fails to train the context encoder, result-  ing in mostly nonsensical representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We presume this is  because the recommended optimizer only takes into account  dmodel and the number of warmup steps, forcing the learning  rate to generally remain much higher than what the learning  rates of the Adam optimizer in regular decay are, preventing  network from convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Results  Visual results including ablations are presented in Fig 4, while  comparison over easy and hard examples are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' There is an obvious improvement over StoryGAN [12],  which fails to generate the proper sequence, and also lacks in  Input  Embedding  FC  Text  Vector  Fully Connected  Sentence  FC  Input  Elementwise  FC  Scalar  product  Image  Vector  Output  Image  Downsampling(a) Left: Ground truth (T=4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Middle: StoryGAN generated frames, low relevance and object quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Right: Ours, baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  (b) Our results without attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Left: Separate Transformer Encoder for G, Dim, Dst, low object relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Middle: Impartial  Encoder (G and Dim gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Right: Impartial encoder (all G, Dim, Dst gradients), mode collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 4: Ablation studies of our framework indicate the power of the Impartial Transformer (G and Dim gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 5: (a) 1st row ground truth, (b) 2nd row generated frames (ours-Impartial Transformer), (c) 3rd row generated frames  (storyGAN) of 3 stories with T=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' From left to right (every 4 images) difficulty of stories increases due to object occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' 4 indicates the optimal usage  of the Impartial transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Even though our implementa-  tion presents satisfactory results when objects are placed in a  distance from each other (Fig 5, left), in cases when objects  are adjacent or overlap, there are some sacrifices to be made:  either semantics -especially shape and material- are not dis-  tinct enough (Fig 5, middle), or objects are ’swallowed’ by  their neighbors (Fig 5, right), which results in low quality se-  mantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The results of human evaluation experiments over  preference are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Results using automated  metrics are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Our framework clearly out-  performs prior efforts [12, 4, 5] according to Clean-FID [25],  LPIPS [26] and SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' We mainly focus on LPIPS metric for  comparison that reflects human perception, where we achieve  16% improvement over prior approaches [12, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Table 1: Human Evaluation preference (averaged results),  Win% = % times our output stories were preferred over [12],  Lose% for vice-versa, Tie% when equally preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Attribute  Win%  Loose%  Tie%  Visual Quality  25  20  55  Consistency  37  32  31  Relevance  32  30  38  Table 2: Average evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  Frame FID↓  Clean-  FID↓  LPIPS↓  SSIM↑  1st  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='94 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
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+page_content='80 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='81  122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='68  ’All’ refers to global results of the Impartial Transformer and is compare  with the global results of [12], [5], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' Results from [5], [4] are obtained by  re-training on CLEVR-SV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' CONCLUSION  In this work, we developed a transformer-inspired framework  for story visualization, aiming to set a new baseline in litera-  ture by achieving improvements according to perceptual met-  rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content=' The usage of the Impartial Transformer demonstrated  promising directions for the evolution of generative models  in the same track, as few -if any- current implementations ex-  ploit a ’forking’ module jointly trained by two adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  As future work we plan to explore the evaluation part of SV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
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+page_content='  [2] Augustus Odena,  Christopher Olah,  and Jonathon  Shlens,  “Conditional image synthesis with auxiliary  classifier gans,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQf-AaO/content/2301.03563v1.pdf'}
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ENE1T4oBgHgl3EQfWgSE/content/tmp_files/2301.03115v1.pdf.txt

@@ -0,0 +1,2161 @@
+Draft version January 10, 2023
+Typeset using LATEX twocolumn style in AASTeX63
+Discovery of Hydrogen Radio Recombination Lines at z = 0.89 towards PKS 1830−211
+Kimberly L. Emig
+,1, ∗ Neeraj Gupta
+,2 Pedro Salas,3 S´ebastien Muller,4 Sergei A. Balashev,5, 6
+Franc¸oise Combes,7 Emmanuel Momjian,8 Yiqing Song,9, 1 Preshanth Jagannathan,8 Partha P. Deka
+,2
+Gyula I. G. J´ozsa,10, 11 Hans-Rainer Kl¨ockner,10 Abhisek Mohapatra,2 Pasquier Noterdaeme,12, 13
+Patrick Petitjean,12 Raghunathan Srianand,2 and Jonah D. Wagenveld10
+1National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
+2Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
+3Green Bank Observatory, 155 Observatory Road, Green Bank, WV 24915, USA
+4Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden
+5Ioffe Institute, Politekhnicheskaya 26, 194021 Saint Petersburg, Russia
+6HSE University, Saint Petersburg, 190121, Russia
+7Observatoire de Paris, Coll`ege de France, PSL University, Sorbonne University, CNRS, LERMA, Paris, France
+8National Radio Astronomy Observatory, 1003 Lopezville Road, Socorro, NM 87801, USA
+9Department of Astronomy, University of Virginia, 530 McCormick Road, Charlottesville, VA 22903, USA
+10Max-Planck Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, 53121 Bonn, Germany
+11Department of Physics and Electronics, Rhodes University, PO Box 94, Makhanda, 6140, South Africa
+12Institut d’Astrophysique de Paris, Sorbonne Universit´e and CNRS, 98bis boulevard Arago, F-75014 Paris, France
+13Franco-Chilean Laboratory for Astronomy, IRL 3386, CNRS and U. de Chile, Casilla 36-D, Santiago, Chile
+(Received ...; Revised ...; Accepted ...)
+Submitted to ApJ
+ABSTRACT
+We report the detection of stimulated hydrogen radio recombination line (RRL) emission from ionized
+gas in a z = 0.89 galaxy using 580–1670 MHz observations from the MeerKAT Absorption Line
+Survey (MALS). The RRL emission originates in a galaxy that intercepts and strongly lenses the radio
+blazar PKS 1830−211 (z = 2.5). This is the second detection of RRLs outside of the local universe
+and the first clearly associated with hydrogen. We detect effective H144α (and H163α) transitions
+at observed frequencies of 1156 (798) MHz by stacking 17 (27) RRLs with 21σ (14σ) significance.
+The RRL emission contains two main velocity components and is coincident in velocity with H i
+21 cm and OH 18 cm absorption. We use the RRL spectral line energy distribution and a Bayesian
+analysis to constrain the density (ne) and the volume-averaged pathlength (ℓ) of the ionized gas.
+We determine log(ne) = 2.0+1.0
+−0.7 cm−3 and log(ℓ) = −0.7+1.1
+−1.1 pc towards the north east (NE) lensed
+image, likely tracing the diffuse thermal phase of the ionized ISM in a thin disk. Towards the south
+west (SW) lensed image, we determine log(ne) = 3.2+0.4
+−1.0 cm−3 and log(ℓ) = −2.7+1.8
+−0.2 pc, tracing
+gas that is more reminiscent of H ii regions. We estimate a star formation (surface density) rate of
+ΣSFR ∼ 0.6 M⊙ yr−1 kpc−2 or SFR ∼ 50 M⊙ yr−1, consistent with a star-forming main sequence
+galaxy of M⋆ ∼ 1011 M⊙. The discovery presented here opens up the possibility of studying ionized
+gas at high redshifts using RRL observations from current and future (e.g., SKA and ngVLA) radio
+facilities.
+Keywords: galaxies: ISM
+Corresponding author: Kimberly L. Emig
+[email protected]
+∗ Jansky Fellow of the National Radio Astronomy Observatory
+1. INTRODUCTION
+Radio recombination lines (RRLs) result from the ra-
+diative de-excitation of electrons at high excitation lev-
+els of atoms. RRLs with frequencies νrest ≲10 GHz in
+arXiv:2301.03115v1  [astro-ph.GA]  8 Jan 2023
+
+ID2
+Emig et al.
+extragalactic sources probe gas with relatively low den-
+sity that can be stimulated by radio continuum (for an
+overview, see Emig 2021), upon which the line emis-
+sion is significantly enhanced compared with the local
+thermodynamic equilibrium (LTE) Boltzmann distribu-
+tion. The physical conditions of the gas strongly influ-
+ence which principal quantum numbers (thus frequen-
+cies) show enhanced emission. Accordingly, the relative
+intensities of RRLs, the so-called spectral line energy
+distribution (SLED), carry information on the temper-
+ature, density, and pathlength of the diffuse gas com-
+ponent (e.g., Shaver 1975; Salgado et al. 2017a; Oonk
+et al. 2017). Stimulated emission has the added benefit
+that its intensity is proportional to the background con-
+tinuum intensity at a given frequency, SRRL ∝ Sbkg cont.
+Therefore, it is conceivable to observe these RRLs at
+cosmological distances wherever bright radio sources are
+present (Shaver 1978). In contrast, RRLs at high radio
+frequencies (≳ 10 GHz) typically trace spontaneous re-
+combination emission (in LTE), making them a great
+direct measure of ionizing photons and therefore star
+formation rates. However, the flux of spontaneous tran-
+sitions falls off with distance (D) and frequency (ν) as
+S ∝ D−2ν−1, limiting the observation of this faint emis-
+sion to galaxies in the nearby universe.
+The very first extragalactic detections of RRLs, in
+M82 and NGC253, found contributions from stimulation
+(Shaver et al. 1977; Seaquist & Bell 1977; Shaver et al.
+1978). In the case of M82, Bell & Seaquist (1978) mod-
+eled the SLED of hydrogen RRLs and showed that the
+increasing intensity of the RRL emission at ν < 10 GHz
+can only result due to stimulation by the large-scale syn-
+chrotron continuum of the galaxy.
+They determined
+that the hydrogen RRLs originate in ionized gas with
+ne ≈ 150 cm−3 and pathlength ℓ ≈ 110 pc. Soon after,
+Churchwell & Shaver (1979) used the Arecibo 300 m
+telescope to search 21 galaxies and active galactic nu-
+clei (AGN) for RRL emission with the 1.4 GHz re-
+ceiver and three AGN with the 430 MHz receiver, with
+the set-up covering just a single RRL transition. They
+did not detect emission with line-to-continuum ratios of
+τRRL > 10−3. To similar sensitivities, Bell et al. (1984)
+used the Effelsberg 100 m dish at 4.8 GHz to search ten
+galaxies without clear success. Bell & Seaquist (1980)
+discovered the H83α and H99α lines at 10.5 GHz and 6.2
+GHz, respectively, in the GHz peaked-spectrum source
+OQ 208 at z = 0.0763, showing that this RRL emission
+could also only arise due to stimulation. These studies
+used narrow bandwidth receivers and were only sensitive
+to one RRL spectral line per observation.
+To date, 8 of the 23 external galaxies with detected
+RRL emission show evidence for stimulated emission by
+z = 0.89
+10
+1
+100
+101
+102
+103
+104
+105
+Density (cm
+3)
+100
+200
+300
+400
+500
+600
+Principal quantum number
+MALS, 0 < z < 3
+6479
+815.9
+242.3
+102.4
+52.45
+30.37
+HRRL Rest Frequency (MHz)
+Figure 1. The principal quantum numbers, n, of Hydrogen
+RRLs at which the stimulated-only emission of gas with a
+given density peaks. The thick black region marks the maxi-
+mum n in the peak of emission, for ionized gas temperatures
+5000 K ≤ Te ≤ 12 000 K, and the thin black region indicates
+the n for which the peak intensity is more than one half of
+the maximum. The shaded blue region shows the parameter
+space covered by MALS for RRL redshifts of 0 ≤ z ≤ 3, while
+the red hatched region shows the coverage for a z = 0.89 ob-
+servation. To calculate the peak optical depths we do not
+take into account an external radiation field (which has a
+minimal effect), but do take into account collisional broad-
+ening of the line profile occurring in dense gas.
+non-thermal emission (for an overview, see Emig 2021)1.
+These were all local (D < 350 Mpc) and mostly star
+forming galaxies, until recently, Emig et al. (2019) used
+the Low Frequency Array to detect stimulated RRLs
+at z = 1.124 with a rest-frame frequency of 284 MHz.
+They argue that the RRL emission most likely arises
+from carbon in an intervening galaxy along the line-of-
+sight to 3C 190. However, they could not clearly discern
+whether the emission was from carbon or hydrogen since
+the lines are separated by 150 km s−1 and have compa-
+rable intensities at those frequencies in the Milky Way
+(e.g., Anantharamaiah 1985).
+The improved sensitiv-
+ity of high-resolution interferometers and the large frac-
+tional bandwidths that enable deeper searches through
+line stacking are making extragalactic RRL detections
+now feasible.
+Furthermore, given that the frequency
+separation between each RRL is unique, the develop-
+ment of new cross-correlation techniques are enabling
+blind searches of RRLs across redshift space (Emig et al.
+2020).
+Current large-bandwidth spectral line surveys, such as
+the MeerKAT Absorption Line Survey (MALS; Gupta
+et al. 2016), the First Large Absorption line Survey
+(FLASH; Allison et al. 2022), the Search for HI Ab-
+1 We
+also
+refer
+the
+reader
+for
+an
+update
+collection
+of
+extragalactic
+RRL
+detections
+at
+www.tinyurl.com/
+DatabaseForExtragalacticRRLs
+
+Radio Recombination Lines at z = 0.89
+3
+sorption with APERTIF (SHARP; e.g., see
+Morganti
+& Oosterloo 2018) – and in the future with the Square
+Kilometer Array (SKA; e.g., Blyth et al. 2015) – can
+observe tens of RRLs simultaneously, opening the way
+for ionized gas studies with optimum sensitivity to gas
+with electron densities of 1 cm−3 ≲ ne ≲ 104 cm−3
+(e.g., see Fig. 1). These surveys can explore RRLs as
+a tracer of (diffuse) ionized gas in external galaxies for
+the first time in a large systematic way and address at
+what level stimulated RRLs are present in galaxies. Ul-
+timately, these RRL observations will bring new insight
+into the evolution of the (ionized) interstellar medium
+of galaxies and the environment of AGN.
+In particular, MALS is carrying out the most sensi-
+tive search to date (σ ∼ 0.6 mJy beam−1 per 6 km s−1
+channel) of H i 21 cm and OH 18 cm absorption lines at
+0 ≲ z ≲ 2 (Gupta et al. 2016). MALS is observing ∼500
+pointings centered on the brightest (S1 GHz > 0.2 Jy)
+radio sources at declination δ ≲ +30◦ (see Gupta et al.
+2022, for the survey footprint), using the MeerKAT
+(Jonas & MeerKAT Team 2016) L band, nominally cov-
+ering 900–1670 MHz, and UHF band, nominally cover-
+ing 580–1015 MHz. Towards each sight line, the sur-
+vey is sensitive to peak RRL line to continuum ratios
+of τRRL = (0.08 − 1) × 10−3 through line stacking,
+reaching the range of optical depths observed from ion-
+ized gas in the Milky Way (e.g., Roshi & Ananthara-
+maiah 2000). For an electron temperature of 8000 K,
+these optical depths convert into emission measures
+102.6 ≳ EM/cm−6 pc ≳ 104.7 for densities 1 cm−3 ≲
+ne ≲ 104 cm−3 (see Fig. 1).
+The first science verification observations of MALS
+(Gupta et al. 2021; Combes et al. 2021) focused on
+the bright (S1 GHz ≈ 11 Jy) z = 2.507 (Lidman et al.
+1999) blazar, PKS 1830−211 (referred to hereafter as
+PKS 1830).
+PKS 1830 has a radio spectral index of
+α1−15 GHz ≈ −0.26, classifying it as a flat spectrum ra-
+dio quasar (Pramesh Rao & Subrahmanyan 1988; Sub-
+rahmanyan et al. 1990), and at lower frequencies it is
+known to have a spectral turnover (Lovell et al. 1996).
+PKS 1830 is strongly gravitationally lensed (Patnaik
+et al. 1993; Nair et al. 1993) by a galaxy at z = 0.89
+(Wiklind & Combes 1996).
+Its morphology reveals
+two compact radio components, referred to as north-
+east (NE) and southwest (SW), approximately 1′′ apart
+and surrounded by a low surface-brightness Einstein ring
+(e.g., Jauncey et al. 1991). While the NE and SW com-
+ponents are images of the blazar core, the ring is mainly
+due to the jet and a bright knot in the jet (Jin et al.
+2003). PKS 1830 is known to be variable, by up to a
+factor of two in radio, on timescales of hours to years
+(e.g., Pramesh Rao & Subrahmanyan 1988; Mart´ı-Vidal
+et al. 2013; Allison et al. 2017; Marti-Vidal & Muller
+2019), and these variations are seen in all three lensed
+components — the NE, SW, and ring. The continuum
+flux is dominated by the NE and SW components at
+least down to 1.4 GHz (Verheijen et al. 2001; Koopmans
+& de Bruyn 2005), where the Einstein ring contributes
+∼1% to the continuum flux (Combes et al. 2021; Pat-
+naik et al. 1993). For an overview image of the system,
+we refer the reader to Nair et al. (1993).
+Two absorption line systems are present along the
+line-of-sight to PKS 1830.
+The lensing galaxy at z =
+0.89 has become an extragalactic-prototype absorption
+system, with the most molecular species detected to date
+(towards the SW image) (e.g., Wiklind & Combes 1996;
+Muller et al. 2011; Tercero et al. 2020). The z = 0.89
+galaxy has been directly imaged with Hubble Space Tele-
+scope and appears to be a barred-spiral (Courbin et al.
+1998, 2002), but remains weak and elusive. From a well-
+constrained lensing model (Nair et al. 1993; Koopmans
+& de Bruyn 2005; Muller et al. 2020; Combes et al.
+2021), the kinematics (vrot ∼ 260 km s−1) and orienta-
+tion also suggest a nearly face-on (i ∼ 25◦) barred-spiral
+galaxy of ∼1011 M⊙. The SW image of the blazar core
+passes through the galaxy at a radius of Rg ∼ 2.4 kpc
+and a central velocity (from spatially unresolved emis-
+sion) at vcen ∼ 0 km s−1, and the NE image passes
+through at Rg ∼ 5.3 kpc and vcen ∼ −150 km s−1.
+The SW image intercepts a spiral arm of the z = 0.89
+galaxy and traces dense (nH2 ∼ 2000 cm−3) molecu-
+lar gas (Wiklind & Combes 1996; Courbin et al. 2002;
+Muller et al. 2013). The gas along the line of sight to the
+NE image arises in the diffuse ISM (Muller et al. 2011)
+and is bright in H i 21 cm and main OH 18 cm absorp-
+tion (Chengalur et al. 1999; Koopmans & de Bruyn 2005;
+Gupta et al. 2021; Combes et al. 2021). No time varia-
+tion is visible in the cm-line spectra from the z = 0.89
+galaxy (Combes et al. 2021), which contrasts strongly
+with the variations detected in the mm-wave absorp-
+tion spectra (Muller & Gu´elin 2008; Muller et al. 2014;
+Schulz et al. 2015). The second absorption system to-
+wards PKS 1830 at z = 0.19 has been seen only in H i
+21 cm absorption so far (Lovell et al. 1996; Allison et al.
+2017; Gupta et al. 2021).
+MALS observations of PKS 1830 verified system per-
+formance and led to the first detection of OH 18 cm
+satellite lines at z = 0.89, which had previously only
+been detected at z ≲ 0.25 (Gupta et al. 2021; Combes
+et al. 2021). In this article, we use these MALS obser-
+vations to search for radio recombination line emission.
+We aim to understand whether RRLs are present and
+detectable and what they can tell us about photoion-
+ized gas in galaxies and AGN. In Sec. 2, we describe
+
+4
+Emig et al.
+the methods used to process the data.
+In Sec. 3, we
+report (i) detections of hydrogen RRLs in both the L
+and UHF bands originating from ionized gas in the z =
+0.89 galaxy, (ii) tests we performed to verify these re-
+sults, and (iii) non-detections at the additional redshifts
+searched. In Sec. 4, we constrain physical conditions of
+the ionized gas by modeling the stimulated RRL emis-
+sion. Finally, we discuss implications of the ionized gas
+measured by the RRLs in Sec. 5 and conclude in Sec. 6.
+In this article, velocities are reported using the helio-
+centric frame, with respect to z = 0.88582 unless stated
+otherwise, and are converted from frequency using the
+relativistic definition. We use ΛCDM cosmology with
+Ωm = 0.29, ΩΛ = 0.71, and Ho = 70 km s−1 Mpc−1, for
+which 1′′ ∼ 7.860 kpc at z = 0.88582.
+2. DATA AND PROCESSING
+2.1. Observations and Data Reduction
+PKS 1830 was observed with the MeerKAT Radio
+Telescope (Jonas & MeerKAT Team 2016) as the first
+science verification target of MALS (Gupta et al. 2016).
+For the work presented here, we used MALS L band
+spectra originally presented by Gupta et al. (2021).
+Hereafter, we refer to this L band dataset obtained on
+2019 December 19 as “Night1”. We also used UHF band
+spectra from the dataset obtained on 2020 July 13 and
+presented in Combes et al. (2021). In addition to these
+previously published datasets, we observed PKS 1830 a
+second time in L band on 2020 September 18 using 59
+antennas, which we will refer to as “Night2” in the ar-
+ticle.
+For both L band observations, the total bandwidth
+of 856 MHz was centered at 1283.987 MHz, cover-
+ing 856–1712 MHz, and split into 32 768 frequency
+channels.
+This delivered a frequency resolution of
+26.123 kHz, which is 6.1 km s−1 at the center of the
+band. For UHF band, the total observable bandwidth
+of 544 MHz covering 544–1088 MHz was also split into
+32 768 frequency channels, providing a channel resolu-
+tion of 16.602 kHz, or 6.1 km s−1 at the center of the
+band, i.e., 815.992 MHz. For all observations the cor-
+relator dump time was 8 s and the data were acquired
+for all four polarization products, labeled as XX, XY,
+YX and YY. We also observed PKS 1934–638 and/or
+3C 286 for flux density scale and bandpass calibrations.
+Since PKS 1830 is a bonafide VLA gain calibrator at
+this spatial resolution, a separate gain calibrator was
+not observed. The total on-source times on PKS 1830
+are: 40 min (L band Night1), 90 min (UHF band) and
+90 min (L band Night2).
+All MALS datasets have been processed using ARTIP,
+the
+Automated
+Radio
+Telescope
+Imaging
+Pipeline
+(Gupta et al. 2021), a Python-based pipeline using tasks
+and tools from the Common Astronomy Software Appli-
+cations (CASA; McMullin et al. 2007; The CASA Team
+et al. 2022). The specific details of the observations, cali-
+bration, and imaging of L and UHF band datasets can be
+found in Gupta et al. (2021) and Combes et al. (2021),
+respectively. The Stokes-I continuum flux densities of
+PKS 1830 obtained from wideband radio continuum im-
+ages in L Night1 and UHF band with robust=0 weight-
+ing are 11.245±0.001 Jy at 1270 MHz and 11.40±0.01 Jy
+at 832 MHz, respectively.
+The radio emission is un-
+resolved in these images with a spatial resolution of
+12.′′9 × 8.′′1 (position angle = −76.◦3) and 17.′′4 × 13.′′1
+(position angle = +69.◦0), respectively (Gupta et al.
+2021; Combes et al. 2021). The quoted uncertainty of
+the flux densities are based on a single 2D Gaussian fit
+to the continuum emission.
+The continuum flux den-
+sity of PKS 1830 obtained using the Night2 dataset is
+S1.27 GHz = 11.86 ± 0.02 Jy.
+This matches with the
+Night1 measurement within the flux density uncertainty
+(∼5%) expected at these low frequencies.
+Therefore,
+throughout the article we use the average flux density
+from the Night1 and Night2 datasets as the representa-
+tive L band flux density, i.e., S1.27 GHz ≈ 11.5 Jy.
+The spectral line data products from ARTIP for RRL
+analysis are continuum-subtracted XX and YY parallel
+hand image cubes obtained with robust=0 weighting.
+We also note that for spectral line processing ARTIP
+splits the L and UHF bands into 15 spectral windows
+(hereafter SPWs) (see Gupta et al. 2021; Combes et al.
+2021, for details of SPW boundaries). The pixel sizes
+for L and UHF band image cubes are 2.′′0 and 3.′′0, re-
+spectively.
+XX and YY spectra were extracted in all
+SPWs from a single pixel at the location of PKS 1830
+determined from the continuum images. The residual
+flux density in each continuum-subtracted spectrum is
+on the order of 0.5% and required additional spectral-
+baseline subtraction.
+Further details of RRL specific spectral line processing
+are provided in the next section. In passing, we note that
+we also made use of a MALS L band dataset of another
+radio source, PKS 1740-517 (hereafter PKS 1740), also
+known as J1744-5144, observed on 2020 September 20
+(two days after Night2 observations of PKS 1830) with
+an on-source time of 56 min. This observation also used
+PKS 1934-638 and 3C 286 for flux density and bandpass
+calibrations, and the unresolved radio source has a flux
+density of ∼7 Jy at 1270 MHz, comparable to PKS 1830.
+We process this dataset following the procedures de-
+scribed above and use the resultant spectra to establish
+the genuineness of the results obtained for PKS 1830.
+
+Radio Recombination Lines at z = 0.89
+5
+2.2. RRL Spectral Processing
+Considering the rest frequencies of RRLs, 38 and 44
+of the α (∆n = 1) recombination lines (per element)
+fall within the MALS L and UHF band coverage, re-
+spectively.2 At z = 0.89, for example, the observations
+cover 31 α recombination lines in L band spanning prin-
+cipal quantum numbers n = 128 − 158 and 36 α recom-
+bination lines in UHF band spanning n = 148 − 183.
+Because line properties of RRLs are correlated over a
+sizeable frequency range, we stacked the spectral lines
+to drive down the noise and increase the signal to noise
+ratio, as is commonly done in Galactic and extragalactic
+RRL observations (Balser 2006; Emig et al. 2020).
+We began spectral processing by identifying the ob-
+served frequency of an RRL and extracting a spectrum
+equivalent to vsys ± 1500 km s−1 centered on the line.
+We selected this velocity chunk and discarded coverage
+outside of it in order to (i) have a sufficient number
+of channels to minimize errors in the estimation of the
+spectral-baseline continuum level (Sault 1994), while (ii)
+ensuring that a low (≤ 5) order polynomial – with an or-
+der determined by minimizing the reduced χ2 – could be
+fit over a well-behaved bandpass. RRLs that fell within
+±1000 km s−1 from the edge of a spectral window were
+excluded from subsequent processing. We flagged chan-
+nels with persistent radio frequency interference (RFI)
+and/or at the HI 21 cm and OH 18 cm line features
+(at relevant redshifts) (see Gupta et al. 2021; Combes
+et al. 2021), and we discarded the full spectral line chunk
+when flagged channels fell within |v| ∼ 500 km s−1 from
+the line center in order to ensure a reliable fit to the
+spectral-baseline. We also flagged spectral line chunks
+that were clearly contaminated by broad-band RFI and
+reliable estimates of the baseline could not be attained.
+After flagging, we next fit a low order (≤ 5) polyno-
+mial to (line-free) channels in each spectral chunk and
+subtracted the fit. For the results presented in Sec. 3,
+we did not impose a line-blank region, except for the
+z = 0.89 stacks. For the z = 0.89 results, we first stacked
+the spectrum without a line-blank region. Based on the
+significant feature we obtained in that spectrum, we set
+the line blank as -230 km s−1 to +55 km s−1.
+Fig. 2 shows the baseline-subtracted spectral chunks,
+as an example, from L band Night2 observations pro-
+cessed for z = 0.89 RRLs.
+The spectral noise has a
+median of 3.4 mJy across the 17 RRL spectral chunks
+used in the stack. The noise properties and number of
+2 We refer to the reader to the CRRLpy module (Salas et al. 2016)
+found at https://github.com/astrofle/CRRLpy for a list of RRL
+line frequencies.
+lines used in the final stacks can be found in Table 1.
+The noise properties are consistent across the bands and
+between parallel hand spectra.
+We next interpolated each spectral chunk to a com-
+mon velocity grid with channel widths of 1 km s−1, in-
+tentionally oversampling the spectral channels to avoid
+artificially smoothing-out spectral features. The spec-
+tral line chunks were then averaged together using the
+inverse noise squared in line-free channels as a weight
+(e.g., Emig et al. 2020).3 We next smoothed the channel
+resolution to 8 km s−1 using a boxcar averaging kernel,
+to better match the lowest resolution achieved at the
+low frequency end of the bands. At this stage, we had
+obtained a single Hnα spectrum for each parallel hand
+XX and YY. Finally, we averaged the XX and YY spec-
+tra to create a Stokes-I spectrum. We chose to combine
+the stacked XX spectrum and the stacked YY spectrum
+rather than creating a Stokes-I spectrum for each line
+before combining polarizations because it (i) resulted in
+better-behaved, i.e., Gaussian-like noise properties and
+(ii) had the benefit of independently examining the dif-
+ferences in the signal detected in two parallel hand spec-
+tra. The latter is particularly useful in distinguishing
+true astrophysical signal from RFI, which is often lin-
+early polarized.
+3. RESULTS
+We applied the spectral processing procedures de-
+scribed in the previous section to PKS 1830 observations
+at the redshifts (i) z = 0 for Galactic emission, (ii)
+z = 0.19259 for the low redshift intervening absorber,
+(iii) z = 0.88582 for the high redshift intervening ab-
+sorber, and (iv) z = 2.507 for the intrinsic redshift of
+PKS 1830. We detect RRL emission from the z = 0.89
+absorber in both Nights of L band and in UHF band ob-
+servations. We report non-detections and upper limits
+at all other redshifts and bands.
+In Fig. 3, we show the L band detections obtained
+from the z = 0.89 absorber.
+We overlay the parallel
+hand spectra, showing that they are consistent within
+the noise in both nights of observations and therefore not
+due to low-level linearly-polarized RFI. We also overlay
+the Stokes-I spectrum obtained from each night of L
+band observations; they are consistent within the noise,
+giving further evidence that the signal is astrophysical
+in nature. Lastly, we averaged L band Stokes-I spec-
+3 We tested combining the spectral chunks with an additional
+weight that depended upon the (inverse) line frequency (raised to
+a power), but this did not significantly change the S/N properties,
+indicating that the line properties are similar and well-correlated
+across the observing bands.
+
+6
+Emig et al.
+1000
+750
+500
+250
+0
+250
+500
+750
+1000
+Velocity (km s
+1)
+0
+40
+80
+120
+160
+200
+240
+280
+320
+360
+400
+440
+480
+520
+560
+600
+Flux density (mJy)
+XX
+YY
+3.0
+3.5
+4.0
+Noise (mJy)
+0
+500
+Integrated Signal 
+(mJy km s
+1)
+0
+5
+Integrated 
+S/N
+128
+132
+133
+134
+135
+136
+137
+145
+149
+150
+151
+152
+153
+155
+156
+158
+128
+132
+133
+134
+135
+136
+137
+145
+149
+150
+151
+152
+153
+155
+156
+158
+128
+132
+133
+134
+135
+136
+137
+145
+149
+150
+151
+152
+153
+155
+156
+158
+Principal Quantum Number, n
+Figure 2. Spectral properties prior to stacking RRL transitions for L band spectra from Night 2. Velocities are shown with
+respect to z = 0.88582 and spectra are shifted along the ordinate for display purposes. The principal quantum number of each
+spectrum is given on the right hand side ordinate. The shaded gray region in the left most panel indicates the line blank region.
+“Noise”, σ, is the rms (outside of the line blank region) per 8 km s−1 channel of the XX (yellow circles) or YY (purple circles)
+spectrum. “Integrated Signal” = ∆vΣN
+i=0 Si, the sum of emission from the channels inside the line blank region. “Integrated
+S/N” = (ΣN
+i=0 Si)/(
+√
+Nσ), the integrated signal divided by the integrated noise; we note that corresponding to negative values
+of Integral Signal, the Integrated S/N is also negative. The vertical dashed line in the three right panels indicates the median
+value. There are no obvious and significant spurious features that could contaminate the RRL spectrum in our final stacking.
+
+Radio Recombination Lines at z = 0.89
+7
+Table 1. Spectral and Line Properties
+Line-of-sight
+z
+Band
+Nlines
+RRL
+noise
+vcenter
+Speak
+FWHM
+�
+SRRL dv
+�
+τ dv
+(mJy)
+(km s−1)
+(mJy)
+(km s−1)
+(mJy km s−1)
+(km s−1)
+PKS 1830-211
+0.88582
+L
+17
+H 144 α
+0.34
+−117.4 ± 5.3
+1.86 ± 0.12
+131 ± 14
+258 ± 33
+−0.045 ± 0.006
+7.7 ± 4.2
+1.54 ± 0.19
+62.3 ± 9.8
+102 ± 20
+−0.018 ± 0.003
+UHF
+27
+H 163 α
+0.57
+−124.4 ± 9.4
+1.59 ± 0.15
+155 ± 24
+262 ± 47
+−0.046 ± 0.008
+7.7
+0.81 ± 0.25
+80 ± 28
+70 ± 25
+−0.012 ± 0.004
+0.0
+L
+17
+H 175 α
+0.36
+...
+...
+...
+< 22.7
+< |0.0020|
+UHF
+28
+H 203 α
+0.41
+...
+...
+...
+< 26.5
+< |0.0023|
+0.19259
+L
+17
+H 166 α
+0.34
+...
+...
+...
+23.6 ± 7.3
+−0.002 1 ± 0.000 6
+UHF
+33
+H 189 α
+0.43
+...
+...
+...
+< 27.2
+< |0.0024|
+2.507
+L
+14
+H 116 α
+0.40
+...
+...
+...
+< 25.5
+< |0.0023|
+UHF
+19
+H 132 α
+0.55
+...
+...
+...
+< 34.9
+< |0.0031|
+PKS 1740-517
+0.88582
+L
+17
+H 142 α
+0.39
+...
+...
+...
+< 24.8
+< |0.004|
+Note— Uncertainties of the line properties are determined from the variance of each parameter as determined by the fit. Nlines is the
+effective number of recombination lines stacked in the final spectrum. “RRL” refers to the effective radio recombination line transition of
+the stacked spectrum. “noise” is the weighted standard deviation of line-free channels in units of mJy per 8 km s−1channel. vcenter is the
+central velocity of the best fit Gaussian and uncertainty. Speak is the peak amplitude of the best fit Gaussian fit. FWHM is the full-width
+half maximum of the best fit Gaussian.
+�
+SRRL dv is the velocity-integrated flux density of the best-fit Gaussian profile, or in the case of
+an upper limit, the reported value is equal to an integrated signal to noise ratio of 3 assuming a line width of 60 km s−1.
+�
+τ dv is the
+velocity-integrated optical depth computed as −
+�
+SRRL/Sc dv where the values used for the continuum flux density, Sc, are described in
+Sec. 3.
+
+8
+Emig et al.
+tra from Nights 1 and 2, producing a spectrum with
+the highest signal-to-noise ratio attainable which we re-
+fer to as the total-I spectrum. The reduction in spec-
+tral noise of the incrementally combined spectra follows
+root N statistics. The total integrated flux density of
+�
+SH144α dv = 337±16 mJy km s−1 with a maximum in-
+tegrated signal-to-noise ratio (S/N) = (ΣN
+i=0 Si)/(
+√
+Nσ)
+of 21 is computed from the total-I spectrum by integrat-
+ing over the N channels covering the velocity range −230
+to 55 km s−1. The effective transition of the total-I spec-
+trum presented in the bottom panel of Fig. 3 is H144α,
+at a rest-frame frequency of 2179.6 MHz and observed
+at 1155.8 MHz. The effective transition is determined
+from the noise-weighted average of the line transitions
+included in the stack.
+The line properties of the ob-
+served emission are consistent with SH136α < 5 mJy
+upper limits to the H136α line at νrest = 2585.7 MHz
+(νobs = 1371.1 MHz) — a transition which is included
+in our stack — obtained by Mohan et al. (2002).
+In addition to (1) multiple nights of observations and
+(2) comparing XX and YY parallel hand spectra, we ver-
+ified additional evidence that the signal is true recom-
+bination line emission by (3) observing an independent
+detection of RRL emission in UHF band (more details
+below), (4) performing jackknife tests, in which one line
+spectrum at a time is omitted from the stacked spec-
+trum, showing that the signal is not dominated by a
+single outlying spectral chunk4, (5) splitting the lines in
+the band into two groups creating two sub-stacks and
+this resulted in consistent line properties, (6) verifying
+a signal is not reproducible by stacking RRLs at other
+redshifts (more details at the end of the Section and
+see Fig. 4), and (7) finding that an RRL spectrum of
+PKS 1740 stacked at z = 0.89 does not systematically
+produce a signal. We made use of additional MALS L
+band observations of PKS 1740 and followed the spectral
+processing procedures described in Sec. 2. We created
+an average RRL spectrum at z = 0.89 with an effec-
+tive transition of H142α and show it in Fig. 4.
+This
+spectrum reached a noise, σ = 0.39 mJy, comparable to
+the PKS 1830 stack. However, no emission or significant
+spectral features are present in the PKS 1740 spectrum,
+and it is consistent with noise.
+This lends additional
+support to the physical and real nature of the emission
+from the z = 0.89 absorber in PKS 1830.
+Two velocity components dominate the L band H144α
+emission from PKS 1830. In the bottom panel of Fig. 3,
+we show the best fit of two Gaussian profiles and the
+4 We also refer the reader to Figure 2, where the noise, integrated
+signal, and integrated signal-to-noise properties also demonstrate
+no single or few outlying spectra dominate.
+2
+0
+2
+2
+0
+2
+2
+0
+2
+Flux Density (mJy)
+600
+400
+200
+0
+200
+400
+600
+Velocity (km s
+1)
+2
+0
+2
+Figure 3. Comparison of RRL stacked spectra in L band at
+z = 0.89. The top three panels compare spectra from parallel
+hands and observing nights. The bottom panel shows (i) the
+final spectral result for the band with Gaussian profiles fit
+to two components and (ii) underneath, the final spectrum
+with the Gaussian profiles subtracted. The vertical dashed
+lines mark the velocities −230 km s−1 and 55 km s−1 within
+which we integrate the signal of the Total component.
+spectrum that results when the two Gaussian profiles
+are subtracted.
+The properties of the Gaussian fits
+are listed in Table 1; the errors of each quantity are
+determined from the variance of each variable as de-
+termined by the fit.
+The H144α component centered
+on −117.4 ± 5.3 km s−1 arises from the NE sight-line
+(and thus we will refer to this velocity component as
+the NE component hereafter). The H144α component
+centered on 7.7 ± 4.2 km s−1 arises from the SW sight-
+line (and thus we will refer to this velocity compo-
+nent as the SW component hereafter).
+In Table 1,
+we also include the integrated optical depth equal to
+�
+τ dv ≈ −
+�
+SRRL/Sc dv computed by letting Sc of each
+component equal half the total continuum flux density
+in the band, Sc ≈ 5.75 Jy. We assume the continuum
+flux is equally split between the two core components
+following Koopmans & de Bruyn (2005) and Combes
+
+Radio Recombination Lines at z = 0.89
+9
+600
+400
+200
+0
+200
+400
+600
+Velocity (km s
+1)
+2
+0
+2
+4
+6
+8
+10
+12
+14
+Flux density (mJy)
+z = 0.0
+z = 0.19
+z = 2.5
+PKS 1740
+ z = 0.89
+L band
+UHF band
+Figure 4. Non-detection RRL spectra in the spectrum of
+PKS 1830 (top three) and in PKS 1740 (bottom). Spectra
+have been given an arbitrary offset in intensity in multiples
+of 4 mJy. Velocities are defined with respect to the labeled
+redshift.
+600
+400
+200
+0
+200
+400
+600
+Velocity (km s
+1)
+1
+0
+1
+2
+3
+Flux Density (mJy)
+HI absorption / -240
+OH absorption / -30
+0
+1×10
+4
+2×10
+4
+Normalized Flux Density
+Figure 5. The H144α spectrum detected in L band overlaid
+by the rescaled H i and OH absorption spectra. The right-
+hand ordinate shows the RRL flux density normalized by the
+continuum.
+et al. (2021) which show this to be the case and that the
+core components dominate the emission at least down
+to 1.4 GHz. Furthermore, ALMA observations measure
+a NE/SW flux density ratio close to one in July 2019
+(Muller et al. 2021).
+Fig. 5 overlays the H144α spectrum with the H i
+21 cm and OH 18 cm absorption spectra that have been
+rescaled by factors of −240 and −30, respectively, in
+flux density units and obtained from the same MALS
+datasets.
+The RRL emission appears to span a simi-
+lar velocity range as these other diffuse gas tracers and
+likewise, is also dominated by two main velocity compo-
+nents.
+The RRL line centroids of the NE and SW compo-
+nents agree within error with the OH 18 cm profiles,
+which have values of −110±3 km s−1 and 6±3 km s−1,
+respectively.
+However, the RRL centroids are signif-
+icantly offset from the dominant H i components at
+∼ −150 km s−1 and 0 km s−1, respectively, albeit the H i
+profile is more complex with perhaps five velocity com-
+ponents best-fitting the observed profile. The variation
+in central velocity with H i could arise due to different
+filling factors, intrinsic distributions, and shapes of the
+continuum with the higher frequency tracers (OH and
+RRLs) finding better agreement.
+The RRL line width of the NE component also agrees
+within error with the OH absorption width, but the
+widths are significantly different for the SW component,
+with the RRL FWHM of 63 ± 10 km s−1 and the OH
+FWHM of 94.2 ± 5 km s−1.
+An estimate of the H i
+width is close to ∼ 100 km s−1 for both components,
+which would be inconsistent with the RRL widths from
+either component. Given the smaller line width of the
+warmer gas traced by the RRLs in the SW, the emis-
+sion may likely have a smaller filling factor, which is to
+say, fewer individual components contribute to the total
+profile. This can be expected since the SW line of sight
+is dominated by dense molecular gas.
+While the two dominant components of H144α emis-
+sion generally agree with the H i and OH profiles, more
+than two velocity components are discernible in the
+higher S/N spectra of H i and OH. For example, OH
+absorption has an additional component centered at
+−211 ± 3 km s−1 with a FWHM of 28 ± 9 km s−1; the
+RRL line profile fit to the NE component encompasses
+some emission in this velocity range. The H i absorp-
+tion spectrum also shows two velocity features that are
+blueshifted with respect to the main ∼ −150 km s−1
+peak.
+In Fig. 6, we show the UHF band detections obtained
+at the z = 0.89 absorber.
+We overlay the parallel
+hand spectra, showing that they are consistent within
+the noise and thus likely not a result of RFI. Inte-
+grating the Stokes-I spectrum over the velocity chan-
+nels from −230 to 55 km s−1 results in
+�
+SH163α dv =
+309 ± 22 mJy km s−1 and a maximum integrated S/N
+of 14. The effective transition of the final spectrum is
+H163α, with a rest-frame frequency of 1504.6 MHz and
+observed at 797.85 MHz.
+The H163α emission arises across a similar velocity
+range as the H144α emission, yet the peak intensities are
+slightly lower. The distinction of two components is less
+obvious in the H163α stack (UHF band), as compared
+with the H144α stack (L band). We fit two Gaussian
+components to the spectrum, fixing the central velocity
+
+10
+Emig et al.
+2
+0
+2
+600
+400
+200
+0
+200
+400
+600
+Velocity (km s
+1)
+4
+2
+0
+2
+Flux Density (mJy)
+Figure 6. RRL stacked spectra in UHF band at z = 0.89.
+The top panel compares spectra from the two parallel hands.
+The bottom panel is the same as in Fig. 3 but for the UHF
+band spectrum.
+of the SW component equal to that obtained from the
+high S/N spectrum in L band, vcen = 7.7 km s−1. The
+best fits are shown in Fig. 6 and the fit parameters are
+listed in Table 1. As in the L band spectrum, we com-
+pute the integrated optical depth by assuming the UHF
+band continuum flux is equally split towards each core
+component, Sc ≈ 5.7 Jy.
+In Fig. 4, we show the L and UHF band stacked spec-
+tra of PKS 1830 at each redshift where we obtained null
+results: z = 0, z = 0.19, and z = 2.5. The spectral prop-
+erties of the non-detections are included in Table 1 and
+the integrated optical depth,
+�
+τ dv ≈ −
+�
+SRRL/Sc dv,
+is computed by letting Sc equal the total continuum
+flux density in the L and UHF bands, respectively.
+Mohan et al. (2002) previously reported a 5σ upper
+limit to the H158α line from the z = 0.19 absorber
+of SH158α < 0.5 mJy. In our L band spectrum of the
+z = 0.19 stack at an effective transition of H166α, we
+reach a spectral noise of 0.34 mJy, and our 3σ upper
+limit to the peak line emission of SH166α < 1.0 mJy
+is consistent with the results obtained by Mohan et al.
+(2002).
+Lastly, we note that carbon RRL emission becomes
+significantly enhanced only at frequencies ≲ 350 MHz,
+and thus we do not expect to detect it at the frequen-
+cies of our observations.
+The detected signal likely
+arises only from hydrogen emission given that it is co-
+incident in velocity with the HI 21 cm and OH 18 cm
+absorption. A 3σ upper limit to the carbon RRLs at
+z = 0.89 in L band is
+�
+SC144α dv < 22.3 mJy km s−1
+and in UHF band is
+�
+SC163α dv < 37.5 mJy km s−1,
+assuming a line width of 60 km s−1. We searched for
+Hβ emission by stacking all available lines in both L
+and UHF band following the procedures described in
+Sec. 2. We report a 3σ upper limit to the Hβ emission
+of
+�
+SH192β dv < 18.3 mJy km s−1 and an in-band ratio
+of peak H192β/H144α < 0.5, where typical in-band β/α
+ratios are ∼ 0.2 (e.g., Salas et al. 2017).
+4. PHYSICAL CONDITIONS OF IONIZED GAS IN
+THE Z=0.89 ABSORBER
+Because PKS 1830 has a relatively low Galactic
+latitude and is behind the Inner Galaxy,
+(ℓ, b
+=
+12.0◦, −5.7◦), it has not yet been feasible to observe ion-
+ized gas tracers at UV through IR wavelengths from the
+z = 0.89 galaxy (e.g., Djorgovski et al. 1992; Courbin
+et al. 1998).
+However, there have been some indica-
+tions for the presence of ionized gas that could be at-
+tributed to the z = 0.89 galaxy. Firstly, the jet emis-
+sion which forms an Einstein ring shows a complete
+turnover at ∼ 300 MHz in its spectral energy distribu-
+tion (SED), which could be due to free-free absorption
+and would not arise from synchrotron self absorption
+(Pramesh Rao & Subrahmanyan 1988; Jauncey et al.
+1991; Jones et al. 1996; Lovell et al. 1996, and for more
+details, see Sec 5.3). Secondly, jet emission in the Ein-
+stein ring (i.e., not the core emission) is strongly po-
+larized (Pramesh Rao & Subrahmanyan 1988; Subrah-
+manyan et al. 1990), indicating ionized plasma lies along
+the lines of sight. However, it is debated whether the
+ionized gas originates in the Milky Way or the z = 0.89
+galaxy and if the blazar core components are free-free
+or synchrotron-self absorbed (Jones et al. 1996; Guirado
+et al. 1999; Mart´ı-Vidal et al. 2015).
+Stimulated hydrogen radio recombination lines pro-
+vide strong constraints on the gas volume density of elec-
+trons and volume-averaged pathlength. In the following
+sections we model the RRL emission to derive physi-
+cal properties. We use these constraints to estimate the
+ionized gas mass per unit area and the ionizing photon
+flux.
+4.1. RRL line width
+The width of recombination lines as a function of fre-
+quency provides insight into the physical properties of
+the emitting gas.
+A Doppler-broadened profile has a
+constant width in velocity units as a function of fre-
+quency, and its Gaussian profile indicates broadening
+from the intrinsic gas particle motions (e.g., due to the
+temperature of the gas or turbulence) or from multi-
+ple emitting regions rotating at different velocities in a
+galaxy. However, collisional and radiation broadening
+
+Radio Recombination Lines at z = 0.89
+11
+create a Lorentzian line profile and have an increasing
+line width towards lower frequency, thereby informing
+on the electron density of the gas or the incident radia-
+tion field strength, respectively.
+For the NE component, widths of FWHMH144α =
+131±14 km s−1 and FWHMH163α = 155±24 km s−1 are
+consistent within error. For the SW component, widths
+of FWHMH144α = 62.3±9.8 km s−1 and FWHMH163α =
+80±28 km s−1 are also consistent within error. This in-
+dicates that the lines are Doppler broadened. Gaussian
+distributions do fit our observed line profiles reasonably
+well (see Figs. 3 and 6).
+Assuming the line widths of each component are equal
+at the two frequencies, the weighted average of the
+FWHM for the NE component is 137 km s−1 and for
+the SW component is 64 km s−1. We then use the width
+at the lowest frequency to put a firm upper limit on the
+gas density, assuming pressure broadening dominates.
+Note, there is no indication to expect an extreme ra-
+diation field that would cause the line to be radiation
+broadened. Salgado et al. (2017b) provide the FWHM
+of a collisionally broadened profile, ∆νcol = nenγ ·10a/π
+where ∆νcol is in units of Hz, ne is in units of cm−3,
+a = −7.386, γ = 4.439 and n is the principal quantum
+level. We place an upper limit of ne ≲ 7900 cm−3 for
+the NE component and ne ≲ 3700 cm−3 for the SW
+component.
+4.2. RRL SLED
+As shown in Shaver (1975) and Shaver (1978), the
+flux density of an optically thin RRL of principal quan-
+tum number, n, and frequency, ν, observed in front of
+a significantly-brighter background radio source of flux
+density Sbkg,ν and which results from stimulated emis-
+sion is given by
+Sn,ν ≈ −τ ∗
+n,ν (bnβ) Sbkg,ν
+(1)
+where τ ∗
+n,ν is the LTE RRL optical depth, bn is the ra-
+tio of the number of hydrogen atoms in level n between
+the non-LTE and the LTE cases, and β characterizes
+the effect of stimulated transitions.
+bn and β, collec-
+tively referred to as departure coefficients, depend on the
+atomic physics of the hydrogen atom, and their product
+is dependent upon electron density, electron tempera-
+ture, the radiation field, and pathlength. Computation
+of the departure coefficients has been thoroughly stud-
+ied since the first observations of hydrogen RRLs (e.g.,
+Shaver 1975; Hummer & Storey 1987; Salgado et al.
+2017a; Prozesky & Smits 2018). Integrating over the line
+profile and expressing the LTE optical depth of the line,
+the integrated optical depth of an α transition (∆n = 1)
+100
+101
+Rest Frequency (GHz)
+10
+2
+dv  (km s
+1)
+NE
+SW
+Total
+100
+Observed Frequency (GHz)
+Figure 7. The integrated optical depth as a function of fre-
+quency for the NE, SW, and Total components. We overlay
+predicted line intensities from RRL models, from 100 model
+combinations that fall within the reported uncertainties and
+chosen at random. We assume a redshift z = 0.89 for the
+conversion between observed and rest frequencies.
+at high principal quantum numbers takes the form
+�
+τn dv ≈ 6.13 × 10−4 km s-1 (bnβ)
+(2)
+�
+EM
+104 cm-6 pc
+� �
+Te
+104 K
+�-2.5 �
+ν
+GHz
+�-1
+where Te is the electron temperature and EM is the
+emission measure equal to EM =
+�
+nenion dℓ for elec-
+tron density ne and ion density nion integrated over the
+pathlength ℓ. Because the departure coefficients at each
+principal quantum number are strongly dependent upon
+density, they are key to breaking the degeneracy between
+density and pathlength in the emission measure.
+In Fig. 7, we plot the RRL SLED using the inte-
+grated optical depths corresponding to the NE and SW
+Gaussian components from from Table 1.
+We also
+plot (and use for analysis in this section), the inte-
+grated optical depth of the Total component computed
+from the velocity-integrated emission over -230 to 55
+km s−1 and setting the continuum flux density, Sc,
+equal to the flux density in each respective band, i.e.
+�
+τH144α = −0.029 ± 0.001 km s−1 and
+�
+τH163α =
+−0.027±0.002 km s−1. While it would be ideal to model
+the two velocity components individually, the large er-
+rors of the Gaussian fit parameters warrant caution. The
+results from the Total component represent an average
+of the two lines-of-sight.
+To model the recombination line emission, we calcu-
+lated the departure coefficients for a range of electron
+densities and electron temperatures using the code and
+
+12
+Emig et al.
+framework described in Salgado et al. (2017a,b). When
+computing bnβ we only consider the effect of the cosmic
+microwave background on the level populations.
+The
+grid in density was sampled at an interval of 0.5 dex
+and the temperature in multiples of 100. We then inter-
+polate over the grid values using a cubic bivariate spline
+for the following analysis.
+We used Bayesian analysis to constrain the posterior
+distribution of the parameters ne, Te, and ℓ, and we
+assume the gas is fully ionized with nion = ne.
+The
+likelihood function describes the comparison of the ob-
+served integrated optical depth with the model (Eq. 2),
+assuming a Gaussian distribution function with disper-
+sion equal to the measurement uncertainty (see Table 1).
+The model is taken to be Eq. 2 in which the values of ν,
+ne, Te, and corresponding bnβ are input, and the path-
+length is left as a free parameter. We used flat priors for
+the parameters expressed in logarithmic scale. We as-
+sumed reasonable ranges of the parameters: for density,
+log(ne) = [−1, 6] cm−3 for the Total component, and
+we used the upper limits derived from the line width
+for the NE and SW components, log(ne) = [−1, 3.4]
+and log(ne) = [−1, 3.6], respectively, in units of cm−3;
+for pathlength, log(ℓ) = [−9, 6] pc; and for tempera-
+ture, we select a strict range of log(Te) = [3, 4.3] K,
+given that theory and observations substantiate tem-
+peratures of photoionized gas to have typical values of
+Te ≈ 6000 − 10 000 K (e.g., Wenger et al. 2019; Tielens
+2005). We also used the prior that the departure coef-
+ficients must be negative and thus the line appears in
+emission. To sample the posterior distribution we used
+an affine-invariant sampler within the package emcee
+(Foreman-Mackey et al. 2013). We used 20 walkers, 105
+iterations, and verified convergence using an autocorre-
+lation time analysis.
+The 2D and 1D marginal posterior distributions of
+the parameters of the Total component are plotted in
+Fig. 8. They show that density is constrained by the
+models to within 0.6 dex with a maximum a posteriori
+value of ne = 500 cm−3 and 68.3% credible interval of
+130 cm−3 ≤ ne ≤ 2000 cm−3, and that electron temper-
+atures are not well constrained. Typical temperatures
+of photoionized gas have values of Te ≈ 6000−10 000 K,
+with variations that are largely metallicity dependent
+(e.g., Shaver et al. 1983; Wenger et al. 2019). The RRL
+modeling shows that typical temperatures are slightly
+more likely than cooler temperatures. The maximum a
+posteriori value of the volume-averaged pathlength for
+the Total component is ℓ = 0.025 pc and the 68.3% cred-
+ible interval is 7.9 × 10−3 pc ≤ ℓ ≤ 0.13 pc. It is reason-
+able to infer that this ionized gas is not distributed in a
+very thin sheet of ℓ ∼ 0.025 pc, but instead has a volume
+ 
+3.2
+3.6
+4.0
+log(Te) 
+1.6
+2.4
+3.2
+4.0
+log(ne) 
+2
+1
+0
+1
+log( ) 
+3.2
+3.6
+4.0
+log(Te) 
+2
+1
+0
+1
+log( ) 
+Figure 8. Corner plot showing the Total component con-
+straints for the electron densities ne in units of cm−3, elec-
+tron temperatures Te in units of K, and pathlength ℓ in
+units of pc. Contours are drawn at 68% and 95% intervals.
+The histograms show the marginalized distributions and the
+shaded region marks the 68% credible intervals.
+Table 2. Constraints on physical conditions. Temper-
+ature is not well constrained within the allowed range
+log(Te) = [3, 4.3] K.
+vcen
+log(ne)
+log(ℓ)
+log(EM)
+(km s−1)
+(cm−3)
+(pc)
+(cm−6 pc)
+NE
+-120
+2.0+1.0
+−0.7
+−0.7+1.1
+−1.1
+4.1+0.8
+−1.2
+SW
+8
+3.2+0.4
+−1.0
+−2.7+1.8
+−0.2
+3.9+0.7
+−1.0
+Total
+(-230–55)
+2.7+0.6
+−0.6
+−1.6+0.7
+−0.5
+4.9+0.4
+−1.7
+filling factor less than unity, and the emission arises from
+multiple, discrete clouds within the galaxy — either as
+ionized clouds, interface layers of molecular clouds, H ii
+regions (for further information see Sec. 5.1), etc.
+In Fig. 9, we show the marginal posterior distributions
+for all three components and each of the parameters,
+electron density (ne), electron temperature (Te), path-
+length (ℓ), and emission measure (EM). The parameter
+constraints are listed in Table 2, with the maximum a
+posteriori values and 68.3% credible intervals. We do
+not list the temperature in Table 2 because it is not well
+constrained by these measurements. We take 100 RRL
+models at random with properties that fall within the
+
+Radio Recombination Lines at z = 0.89
+13
+100
+102
+104
+Density (cm
+3)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Posterior Probability
+NE
+SW
+Total
+1000
+10000
+Temperature (K)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+10
+2
+100
+102
+Pathlength (pc)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+102
+104
+106
+Emission Measure (cm
+6 pc)
+0.0
+0.1
+0.2
+0.3
+0.4
+Figure 9.
+Posterior probability distributions for the parameters constrained through the RRL observations: electron density ne,
+electron temperature Te, pathlength ℓ, and emission measure EM, derived for the NE, SW, and Total components. The shaded
+regions represent the 68.3% credible intervals. In the left-most panel, the gray dashed (dotted) line marks the ne < 7900 cm−3
+(<3700 cm−3) constraint for the NE (SW) component obtained from the line width.
+constraints of the 68.3% credible intervals and plot the
+SLEDs of these models in Fig. 7, in order to demonstrate
+the variation in the predicted integrated optical depth
+for different models.
+4.3. Mass of ionized gas
+Using the most likely values of the physical quantities
+in Table 2, we estimate the mass and mass per unit area
+of ionized gas in each component. These mass estimates
+are meant for qualitative comparisons as order of magni-
+tude indications and do warrant caution. We assume the
+volume of gas is effectively described by a cylinder, as
+a circular region on the plane of the sky (core size) and
+with a distance into the plane equal to the path length.
+We also assume that the radio continuum emission is
+dominated by the NE and SW components, as these
+have been shown to account for ∼ 99% of the emission
+at 1.4 GHz (Koopmans & de Bruyn 2005; Combes et al.
+2021). To calculate the mass of ionized gas, we expect
+Mion ≈ 1.36mH · ne · (πr2
+c)ℓ
+Mion ≈ 0.11 M⊙
+� ne
+cm−3
+� � rc
+pc
+�2 � ℓ
+pc
+�
+.
+(3)
+where mH is the hydrogen atom mass and rc is the radius
+of the radio continuum core. Guirado et al. (1999) con-
+strained the source size of the SW component to follow
+∝ ν−2.0 resulting in a size of 0.1′′ at 1 GHz, which cor-
+responds to the value we set of rc = 786 pc at z = 0.89.
+For the NE component we adopt the separation of 0.05′′
+between the two brightest emission peaks as the size,
+rc = 393 pc.
+We do not include uncertainties for rc
+when computing the mass; however, we note that their
+errors are small in comparison to the large range un-
+certainty of the credible intervals. Using the posterior
+distributions of ne and ℓ, we estimate a total mass for
+the NE component of Mion ≈ 106.0+0.6
+−0.7 M⊙ and for the
+SW component, Mion ≈ 105.5+0.8
+−0.4 M⊙. If we assume that
+the area of the Total component is a summation of the
+areas intercepted by the NE and SW lines-of-sight, the
+estimated ionized mass is Mion ≈ 106.4+0.7
+−0.5 M⊙.
+It is informative to also calculate the gas mass per
+unit area
+Σion ≈ 0.033 M⊙ pc−2 � ne
+cm−3
+� � ℓ
+pc
+�
+.
+(4)
+For the NE, SW, and Total components, we calculate
+Σion to be 100.3+0.6
+−0.7 M⊙ pc−2, 10−0.8+0.8
+−0.4 M⊙ pc−2, and
+100.0+0.7
+−0.5 M⊙ pc−2, respectively.
+Using the surface densities of the NE, SW and Total
+components, we calculate an estimate for the total ion-
+ized gas mass of the galaxy within the Einstein ring of
+Rg ∼ 5.3 kpc in units of M⊙ as log(Mion,g) ≈ 8.2+0.6
+−0.7,
+7.1+0.8
+−0.4, and 7.9+0.7
+−0.5, respectively. While the mass esti-
+mated from the SW is more equivalent to a true lower
+limit of the total ionized gas mass, the estimates from
+the NE and Total components are almost certainly lower
+limits as well since we do not trace the bulk of the ionized
+gas mass of the galaxy contained in the Warm Ionized
+Medium (WIM; see Tielens 2005).
+4.4. Ionizing photon flux
+We use the ionized gas emission measure to infer the
+ionizing photon flux. The ionizing photon rate, Qo, per
+unit area is,
+Qo
+area ≈ EM · αB
+Qo
+area ≈ 7.6 × 1045 photons s−1 pc−2
+EM
+103 cm−6 pc (5)
+where αB is the case B recombination coefficient. Us-
+ing the posterior distributions of the emission mea-
+sure, we calculate ionizing photon fluxes in units of
+
+14
+Emig et al.
+photons s−1 pc−2 of log (Qo/area) = 47.0+0.8
+−1.2, 46.8+0.7
+−1.0,
+and 47.8+0.4
+−1.7 for the NE, SW, and Total components,
+respectively. These values are about an order of mag-
+nitude or more higher than the ionizing photon flux in
+the disk of the Milky Way (Kado-Fong et al. 2020, and
+references therein).
+While the gas mass estimates (Sec. 4.3) are likely lower
+limits, the ionized photon fluxes are closer to realistic
+values, since they are dominated by the large emission
+measures that we are sensitive to.
+5. DISCUSSION
+5.1. Star Formation Rate and ISM Properties
+The posterior distributions for the emission mea-
+sure and thus directly the ionizing photon flux are
+constrained with fairly similar 68.3% credible inter-
+vals.
+To estimate a star formation rate (SFR) of
+the galaxy, we adopt the log-average EM within the
+credible intervals of the three components, EM
+≈
+104.0±0.8 cm−6 pc, within a typical error of 0.8 dex, and
+therefore we adopt an ionizing photon flux is Qo/area ≈
+1046.9±0.8 photons s−1 pc−2. We estimate the total ion-
+izing photon rate for the galaxy within Rg = 5.3 kpc
+as Qo = πR2
+g · 1046.9±0.8 ≈ 1054.8±0.8 photons s−1. A
+Starburst99 model (Leitherer et al. 1999) of continu-
+ous star formation establishes the relation with the ion-
+izing photon rate (which levels off after ∼50 Myr) of
+SFR = 1 M⊙ yr−1 (Qo/1.4 × 1053 photons s−1). Using
+this relation, an estimated SFR for the z = 0.89 galaxy
+is SFR ≈ 101.7±0.8 M⊙ yr−1 and the SFR per unit area
+is ΣSFR ≈ 10−0.2±0.8 M⊙ yr−1 kpc−2. A galaxy on the
+main sequence at z = 0.89 with SFR ∼50 M⊙ yr−1 has a
+typical stellar mass of ∼ 1011 M⊙ (e.g., Schreiber et al.
+2015). Current lensing models estimate the total mass
+within the Einstein ring as ME ≈ 4 × 1011 M⊙ (Muller
+et al. 2020) and therefore a stellar mass of ∼ 8×1010 M⊙
+given a typical mass to stellar light ratio of ∼ 5 (Treu &
+Koopmans 2004). This is likely a main-sequence galaxy.
+In Sec. 4.3 we estimated the ionized gas mass per unit
+area of the NE component to be Σion ∼ 2.1 M⊙ pc−2.
+For comparison, the H i column density is estimated to
+be NH I ≈ 5×1021 cm−2 assuming half of the continuum
+flux comes from the NE component (Chengalur et al.
+1999; Combes et al. 2021). Assuming the average parti-
+cle mass is ≈ 1.36mH, the atomic gas mass per unit area
+is ΣH I ≈ 50 M⊙ pc−2. We use the OH column density of
+NOH ≈ 1.5 × 1015 cm−2 (Gupta et al. 2021) and assume
+an abundance of 10−7 (Balashev et al. 2021) given that
+this line-of-sight has properties of a diffuse cloud (Muller
+et al. 2011) in order to estimate a molecular gas column
+density of NH2 ∼ 1.5×1022 cm−2. This H2 column den-
+sity is higher than the NH2 ∼ 1021 cm−2 derived from
+H2O absorption, with a smaller continuum cross section
+at higher frequencies (Muller et al. 2014). Assuming the
+average particle mass is ≈ 2mH, the molecular gas mass
+per unit area is ΣH2 ≈ 240 M⊙ pc−2.
+With these estimates, the neutral gas mass per unit
+area is ΣHI+H2
+∼ 290 M⊙ pc−2.
+The ionized gas
+mass detected in RRLs of the NE component is a
+small fraction (∼0.7%) of the total gas mass.
+The
+Kennicutt-Schmidt law (Kennicutt 1998) long estab-
+lishes a direct correlation between the surface densi-
+ties of the neutral gas mass ΣHI+H2 and the star for-
+mation rate ΣSFR in galaxies on spatial scales 300–500
+pc or more (Schruba et al. 2010; Kruijssen & Long-
+more 2014). It suggests that a region in a galaxy with
+ΣSFR ∼ 0.6 M⊙ yr−1 kpc−2 has a neutral gas mass per
+unit area of ΣHI+H2 ∼ 270 M⊙ pc−2 (e.g., Kennicutt
+& De Los Reyes 2021). Our measured neutral gas mass
+agrees to within 10% of the expected value, although our
+measured SFR surface density has a large uncertainty.
+Although the estimated densities of the NE, SW
+and Total components are consistent within the er-
+rors, the most likely density of the SW component,
+ne ≈ 1600 cm−3, is typical of young, compact H ii re-
+gions. If we assume the H ii regions are rH II ≈ 2 pc in
+size, then the covering fraction through this cross section
+of the galaxy is fc ≈ ℓSW/(4/3 · rH II) ≈ 7.5 × 10−4. The
+total number of H ii regions, NH II, is calculated from
+the surface density estimates, πr2
+C = NH II/fc · (π r2
+H II),
+which computes to NH II ∼ 116, where we recall from
+Sec. 4.3 that rC = 786 pc for the SW component. Equiv-
+alently, assuming all H ii regions have ne ≈ 1600 cm−3
+and rH II ≈ 2 pc, the ionized gas mass per H ii region is
+estimated to be MH II = 1860 M⊙, and from the total
+ionized mass of the SW component, MSW = NH IIMH II,
+also results in NH II ≈ (2.1×105 M⊙)/(1860 M⊙) ∼ 113.
+However, this calculation warrants caution because the
+number of H ii regions changes dramatically depending
+on the assumed size. We set rH II ≈ 2 pc because it is
+the typical size of young, massive star clusters (Ryon
+et al. 2017) that are expected to dominate star forma-
+tion in galaxies of this epoch, i.e., with high gas surface
+densities and star formation rates (Kruijssen 2012).
+5.2. The ne − ΣSFR relation
+The star formation rate per unit area is shown to
+be correlated with the electron density of ionized gas
+in the region (Shimakawa et al. 2015; Herrera-Camus
+et al. 2016; Jiang et al. 2019). Assuming H ii regions
+thermalize with the ISM (Guti´errez & Beckman 2010),
+i.e., take on a pressure balance with others phases, then
+the volume-average density of the ionized gas (along
+with fairly consistent ionized gas temperatures) indi-
+
+Radio Recombination Lines at z = 0.89
+15
+cates the thermal pressure of the medium (Jiang et al.
+2019; Barnes et al. 2021). This results in a P −ΣSFR re-
+lation that serves as an important test-bed for pressure-
+regulated, feedback-modulated star formation (e.g., Kim
+et al. 2013; Ostriker & Kim 2022). Using doublet ratios
+of [SII] and [OII] in the optical (Kewley et al. 2019),
+Shimakawa et al. (2015) and Jiang et al. (2019) find a
+relation of ΣSFR ∝ n1.7±0.3
+e
+in a sample of z ∼ 1 − 3
+starburst galaxies. In the far IR, the ratio of [NII] fine
+structure lines from a sample of nearby normal galax-
+ies and (ultra) luminous infrared galaxies ((U)LIRGs;
+LIR ≥ 1011 L⊙) establish ΣSFR ∝ n1.5
+e
+(Herrera-Camus
+et al. 2016).
+For ΣSFR ∼ 0.6 M⊙ yr−1 kpc−2, the electron den-
+sity predicted by the Shimakawa et al. (2015) relation is
+∼ 110 cm−3 and by the Herrera-Camus et al. (2016) re-
+lation is ∼ 200 cm−3.
+These density estimates agree
+well with the NE component and lie on the cusp of
+the 68% credible intervals of the SW and Total com-
+ponents.
+If the RRL emission is tracing the thermal
+properties of the diffuse medium in the galaxy’s disk,
+higher pressures and densities are typically found at
+smaller galactic radii, i.e., the SW component, than at
+larger galactic radii, i.e., the NE component. For exam-
+ple, Guti´errez & Beckman (2010) measured the electron
+density to increase towards smaller galactic radii, r, as
+⟨ne⟩ = ⟨ne⟩o exp (−r/Rg), where Rg is the scale length
+of the disk at which star-formation and density drops off
+and ⟨ne⟩o is inner most density. With Rg = 5.3 kpc and
+⟨ne⟩ = 100 cm−3 of the NE component, at r = 2.4 kpc
+the expected density is ne ∼ 160 cm−3. While the con-
+trast in density at the two radii is not as extreme as the
+best fit densities of the components indicate, the general
+trend is consistent and a density of 160 cm−3 does fall
+within the 68% credible interval of the SW component.
+The smaller pathlength in comparison to the NE com-
+ponent might then indicate a smaller covering fraction
+of the overall larger cross section of this line-of-sight, i.e.
+only within or close to the spiral arm.
+5.3. Radio Continuum SED
+The radio continuum emission from PKS 1830 is com-
+plex. The lensing is achromatic and contains a small-
+scale core-jet structure with regions of different spectral
+indices and opacities. In addition, the source is variable
+on hourly to yearly timescales (Pramesh Rao & Sub-
+rahmanyan 1988; van Ommen et al. 1995; Lovell et al.
+1996, 1998; Garrett et al. 1997; Jin et al. 2003; Mart´ı-
+Vidal et al. 2013; Allison et al. 2017), which makes it
+difficult to compare observations and model the radio
+continuum emission (Muller et al. 2020).
+10
+1
+100
+101
+Rest Frequency (GHz)
+101
+3 × 100
+4 × 100
+6 × 100
+2 × 101
+SC (Jy)
+10
+1
+100
+101
+Observed Frequency (GHz)
+Figure 10.
+Compilation of continuum flux density mea-
+surements from the bright and highly variable (factors of
+∼1.5 on weeks and years timescales) PKS 1830. The gray
+shaded region encompasses the 1σ confidence region of a
+best-fitting power-law that is attenuated by free-free absorp-
+tion. We also include the continuum SEDS (with colors cor-
+responding to the components on the left hand plot) from a
+fixed power-law and that is attenuated by ionized gas with
+properties constrained by the RRL models.
+We assume a
+redshift z = 0.89 for the conversion between observed and
+rest frequencies.
+Ionized gas that is detectable in RRLs and has a
+large covering fraction would also emit free-free emis-
+sion and when it becomes optically thick, would ab-
+sorb any background radio continuum.
+A reliable
+model and knowledge of the spatially-resolved radio
+SED could independently constrain the physical con-
+ditions of the gas through free-free absorption.
+The
+frequency at which radio emission becomes optically
+thick to free-free absorption is defined by τν = 6.67 ×
+10−2 EM T −1.323
+e
+(ν/GHz)−2.118 (e.g., Emig et al. 2022).
+We
+collected
+radio
+continuum
+measurements
+of
+PKS 1830 at ν ≲ 22 GHz (using the NASA/IPAC
+Extragalactic Database (NED) and Pramesh Rao &
+Subrahmanyan 1988; Henkel et al. 2008; Intema et al.
+2017) and plot these in Fig. 10.
+We fit the SED of
+PKS 1830 as a power-law index with an external screen
+of free-free absorption, using the functional form Sν =
+So( ν
+νo )α exp(−τν) with τν = τo( ν
+νo )−2.118. Setting νo =
+40 GHz with respect to observed frequencies, the best fit
+parameters with 1σ uncertainties are So = 4.8 ± 0.4 Jy,
+α = −0.24 ± 0.03 and τo = (1.1 ± 0.2) × 10−5.
+In Fig. 10, we also plot how ionized gas that has phys-
+ical properties constrained by the RRL models — us-
+ing the same selection of models presented in Fig. 7 —
+would attenuate a power-law continuum SED (normal-
+ized by So and α of our fit). The emission from the RRLs
+
+16
+Emig et al.
+we detect at z = 0.89 would result in free-free absorp-
+tion at lower frequencies than is observed in PKS 1830.
+For the z = 0.89 galaxy to cause the free-free absorp-
+tion, volume-average pathlengths a factor of 5 larger are
+needed. A smaller filling factor and larger pathlength
+intercepting the radio emission is not unreasonable.
+It would still be possible for ionized gas in the envi-
+ronment of the blazar at z = 2.5 to create the absorp-
+tion in the radio SED. Even though we do not detect
+hydrogen RRL emission at z = 2.5, ionized gas with dif-
+ferent properties, for example higher densities, could be
+present and still be consistent with the RRL constraints.
+We also note that ionized gas in the Milky Way with
+EMs that result in a turnover at the observed frequen-
+cies (especially, on < 1′′ scales) would be observable in
+hydrogen RRL emission at z = 0, and we do not detect
+any RRL emission from the Milky Way.
+6. CONCLUSIONS
+We used MALS observations to detect RRL emission
+in the spectrum of the radio blazar PKS 1830. The RRL
+emission is observed at z = 0.89 from a galaxy that
+lies along the line of sight and strongly lenses PKS 1830.
+This is the second detection of RRLs outside of the local
+universe (i.e., at z ≥ 0.076) and the first clearly associ-
+ated with hydrogen (e.g., Emig et al. 2019). We detect
+H144α by stacking 17 RRLs covered by the L band (856–
+1712 MHz) with a S/N of 21 (see Fig. 3), and we detect
+H163α by stacking 27 lines in the UHF band (544–1088
+MHz) with a S/N of 14 (see Fig. 6). Emission from the
+H144α line is consistent over two separate observations,
+when comparing the spectra in parallel hand polariza-
+tions, and is robust against additional spectral stacking
+verification methods. Like the H i 21 cm and OH 18 cm
+absorption spectra (see Fig. 5), the H144α and H163α
+emission profiles span ∼250 km s−1 in velocity and are
+dominated by two velocity components associated with
+two physically distinct regions of the galaxy, the NE and
+SW lines of sight. We do not detect RRL emission in
+either band intrinsic to PKS 1830 (z = 2.5), from the
+z = 0.19 absorption system along this line of sight, or
+from the Milky Way (see Fig. 4).
+Hydrogen RRL emission typically arises from fully
+ionized gas and only stimulated emission is observable
+outside of the local universe.
+The maser-like proper-
+ties of stimulated emission enable the RRL SLED to
+constrain the density and pathlength of the ionized gas
+(see Table 2 and Fig. 9).
+Considering the total inte-
+grated line intensity, referred to as the Total compo-
+nent, we used a Bayesian analysis to constrain the elec-
+tron density of the gas log(ne) = 2.6 ± 0.6 cm−3 and
+a volume-averaged pathlength of log(ℓ) = −1.6+0.7
+−0.5 pc,
+which likely has a non-unity filling factor.
+Analyzed
+separately, the NE line-of-sight appears to harbor less
+dense gas with log(ne) = 2.0+1.9
+−0.7 cm−3 and log(ℓ) =
+−0.7±1.1 pc, and the SW line-of-sight appears to inter-
+cept dense gas that is more typical of H ii regions with
+log(ne) = 3.2+0.4
+−1.0 cm−3 and log(ℓ) = −2.7+1.8
+−0.2 pc. These
+scenarios are consistent with the NE line-of-sight pass-
+ing through diffuse clouds at a larger galactic radius, and
+the SW component directly intercepting a spiral arm, as
+has previously been determined.
+The RRL components measure an ionizing pho-
+ton flux of Qo/area
+≈
+1046±0.8
+photons s−1 pc−2
+and star formation rate surface density of ΣSFR ∼
+10−0.2±0.8 M⊙ yr−1 kpc−2.
+Taken over the z = 0.89
+galaxy within Rg ∼ 5.3 kpc, the ionizing photon rate of
+Qo ∼ 1054.8 photons s−1 yields an average star forma-
+tion rate of SFR ∼ 50 M⊙ yr−1. Despite the plethora of
+molecular species observed, the ionized gas content and
+SFR have not been previously measured for this source,
+largely due to the highly reddened nature of PKS 1830
+at optical and NIR wavelengths. In comparing the SFR
+and the galaxy’s mass (from lensing), the z = 0.89 sys-
+tem is likely on the main sequence.
+The ionized gas mass per unit area of the diffuse
+NE component as measured by the RRL emission is
+Σion ≈ 2.1 M⊙ pc−2, in comparison with gas masses of
+ΣH I ≈ 50 M⊙ pc−2 and ΣH2 ≈ 240 M⊙ pc−2 (via OH)
+estimated from only the MALS observations. Given our
+estimated SFR, the H i+H2 gas mass surface density
+is close to the gas content predicted by the Kennicutt-
+Schmidt law.
+Our measured electron densities also
+match reasonably well with the ne − ΣSFR relation de-
+termined from optical and FIR line ratios.
+PKS 1830 is the first source investigated with MALS,
+and the detection of RRLs in the source is promising
+for the remaining ∼500 targets of the survey. With the
+first hydrogen RRL detection that breaks the redshift-
+barrier, we show that this tracer can be an important
+tool for investigating (a) the electron density (thermal
+pressure) of ionized gas in the ISM of galaxies (and the
+ne−ΣSFR relation) and (b) the SED of AGN, thus even-
+tually AGN evolution. We have also demonstrated the
+unique science that can be achieved through H i 21 cm,
+OH 18 cm, and RRL measurements that are simulta-
+neously observed in the MALS survey. The ionized gas
+properties in the z = 0.89 galaxy will be substantially
+improved through RRL observations at higher and lower
+radio frequencies and at higher (<1′′) spatial resolutions
+which can separate the two main (velocity) components
+of emission. The new science afforded by high-redshift
+RRL studies is accessible with on-going wide-bandwidth
+spectral line surveys and will be explored in unprece-
+
+Radio Recombination Lines at z = 0.89
+17
+dented capacities with future facilities such as the next
+generation Very Large Array (ngVLA; Murphy et al.
+2018) and the SKA (Carilli 2015).
+ACKNOWLEDGMENTS
+The authors acknowledge and appreciate the efforts
+and input of the anonymous reviewer of this article. The
+authors thank Peter Shaver for comments on the article
+and for the inspiration and motivation to carry through
+with this research.
+SAB was supported by RSF grant 18-12-00301. The
+MeerKAT telescope is operated by the South African
+Radio Astronomy Observatory, which is a facility of the
+National Research Foundation, an agency of the De-
+partment of Science and Innovation.
+The MeerKAT
+data were processed using the MALS computing facil-
+ity at IUCAA (https://mals.iucaa.in/releases) The Na-
+tional Radio Astronomy Observatory is a facility of the
+National Science Foundation operated under coopera-
+tive agreement by Associated Universities, Inc.
+This
+research has made use of the NASA/IPAC Extragalac-
+tic Database (NED), which is funded by the National
+Aeronautics and Space Administration and operated by
+the California Institute of Technology.
+Facilities: MeerKAT
+Software: ARTIP (Gupta et al. 2021), Astropy (The
+Astropy Collaboration 2018; The Astropy Collaboration
+et al. 2022), CASA (McMullin et al. 2007; The CASA
+Team et al. 2022), ChainConsumer (Hinton 2016), CR-
+RLpy (Salas et al. 2016), emcee (Foreman-Mackey et al.
+2013), Matplotlib (Hunter 2007), and NumPy (Harris
+et al. 2020)
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F9FLT4oBgHgl3EQfGi9H/content/tmp_files/2301.11992v1.pdf.txt

@@ -0,0 +1,3004 @@
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL
+COMPLEXITY
+J. D¨OLZ
+Abstract. We consider the H2-formatted compression and computational estimation of co-
+variance functions on a compact set in Rd. The classical sample covariance or Monte Carlo
+estimator is prohibitively expensive for many practically relevant problems, where often ap-
+proximation spaces with many degrees of freedom and many samples for the estimator are
+needed. In this article, we propose and analyze a data sparse multilevel sample covariance es-
+timator, i.e., a multilevel Monte Carlo estimator. For this purpose, we generalize the notion of
+asymptotically smooth kernel functions to a Gevrey type class of kernels for which we derive
+new variable-order H2-approximation rates. These variable-order H2-approximations can be
+considered as a variant of hp-approximations. Our multilevel sample covariance estimator then
+uses an approximate multilevel hierarchy of variable-order H2-approximations to compress the
+sample covariances on each level. The non-nestedness of the different levels makes the reduction
+to the final estimator nontrivial and we present a suitable algorithm which can handle this task
+in linear complexity. This allows for a data sparse multilevel estimator of Gevrey covariance ker-
+nel functions in the best possible complexity for Monte Carlo type multilevel estimators, which
+is quadratic. Numerical examples which estimate covariance matrices with tens of billions of
+entries are presented.
+1. Introduction
+1.1. Motivation. Covariance functions or kernel functions
+g: D × D → R,
+on a compact set D ⊂ Rd arise in many fields of application such as Gaussian process computations
+[44], machine learning [33, 49], and uncertainty quantification [23]. However, in many cases these
+functions are not available in closed form, but must be suitably estimated from samples. The
+canonical estimator for this purpose is the sample covariance estimator or Monte Carlo estimator
+g ≈ 1
+M
+M
+�
+k=1
+z(k) ⊗ z(k),
+see, e.g., [34], where the sample functions z(k), k = 1, . . . , M, are assumed to be independent,
+identically distributed (i.i.d.) elements of a Hilbert space and ⊗ is understood as the Hilbertian
+tensor product.
+The challenge with the above estimator is that the covariance function and
+the samples are often infinite-dimensional objects which in practice need to be discretized for
+computational purposes. After discretization, the sample functions themselves are represented as
+elements of Rn and the covariance function as a covariance matrix in Rn×n. Assuming that the
+samples are approximated to an accuracy of ε = n−α, roughly M = ε−2 = n2α samples need
+to be drawn to reach an overall error of O(ε) of the sample covariance estimator.
+Thus, the
+computational effort of the sample covariance estimator is O(Mn2) = O(ε−2−2/α) = O(n2α+2).
+This is prohibitive for large n, as it is often required for sufficient accuracy in applications.
+This article presents an algorithm with rigorous error bounds for approximating the covariance
+function in optimal complexity.
+Here, optimal complexity is understood such that estimating
+the covariance has asymptotically the same complexity as estimating the mean, i.e., as good as
+O(ε−2) = O(n2α) to reach an accuracy of O(ε) under certain assumptions on the underlying
+approximation space.
+1
+arXiv:2301.11992v1  [math.NA]  27 Jan 2023
+
+2
+J. D ¨OLZ
+1.2. Related work. The challenges of large covariance matrices are commonly overcome by using
+data sparse approximations. Here, the main difference between methods is how the data sparse
+format is chosen. Purely algebraic methods operate in a black-box fashion on the samples of the
+sample covariance estimator to estimate suitable compression parameters for previously chosen
+data sparse formats such as banded matrices [3] or sparse matrices [2, 3, 19, 20, 21]. See also
+[11] for recent literature review.
+However, a simultaneous estimate on approximation quality
+and computational complexity is not available without additional assumptions on the algebraic
+properties of the samples and/or covariance matrix. These properties are usually inferred from
+assumed analytical properties of the underlying statistical model. Here, an often considered analog
+to some of the matrix approximation classes considered in [2] are asymptotically smooth covariance
+functions, which assume a certain decrease of the covariance with increasing spatial distance.
+These kinds of functions are also considered in the fast multipole method [25] and its and abstract
+counterparts H- and H2-matrices [4, 27], as well as in wavelet compression [47]. The first have been
+applied in machine learning [5] and uncertainty quantification [18, 30, 35, 48] where complexity and
+approximation estimates have been derived. The available machinery was also applied to estimate
+hyperparameters of covariance functions [12, 22, 36, 39, 41], but we stress that the objective of
+this article is to estimate the full covariance functions. Finally, wavelet based approaches have
+been used in [28, 29, 30, 32, 46] for compression and estimation of covariance functions. Similar
+to wavelet based approaches, sparse grid approaches are also based on a multilevel hierarchy and
+provide a sparse representation of the covariance matrix, but assume some global smoothness of
+the covariance [1, 10]. All of the mentioned methods operating on assumed analytical properties
+of covariance functions are capable to reduce the storage requirements of corresponding covariance
+matrices in Rn×n from O(n2) to O(n) or O(n logβ n), β > 0, with a negligible approximation error.
+Thus, the n2 part of the computational cost of the sample covariance estimator can significantly
+be reduced.
+Reducing the computational cost of the sampling process can essentially achieved by two ap-
+proaches. The first approach is to see the sample covariance estimator as a Monte Carlo quadrature
+for a stochastic integral and to replace that quadrature rule by a more efficient method such as
+quasi-Monte Carlo methods [16] and sparse grid approaches [10]. However, bare strong assump-
+tions, further measures to reduce the number of samples are required. The second approach to
+reduce computational cost during sampling are variance reduction techniques and in particular
+the multilevel Monte Carlo method, see, e.g., [24, 31] for a general overview. The basic idea is
+to exploit a multi-level hierarchy in the approximation spaces for the covariance discretization to
+obtain covariance matrices of decreasing size and to combine many smaller and only a few larger
+matrices to a covariance estimator. It was applied to smaller and dense covariance matrices in [42]
+for the estimation of Sobol indices and to larger covariance matrices combined with a sparse grid
+approximation in [1, 14] and combined with a wavelet approximation in [28].
+1.3. Gδ-asymptotical smoothness and Gevrey kernels. As we will show in a moment, there
+is a large class of covariance functions which is not asymptotically smooth. The first objective of
+this paper is to generalize some of the available H2-compression techniques, which can be seen as
+a special variant of hp-approximation, to a more general class of covariance functions. However,
+we stress that all of the presented algorithms also apply to the classical, asymptotically smooth
+kernel functions.
+To this end, we assume that D is equipped with a measure µ, write L2
+µ(D) = L2(D), and
+assume that we are given a probability space (Ω, Σ, P). Following the stochastic partial differential
+equation approach to Gaussian random fields [38, 51], we note that realizations Z ∈ L2
+P(Ω; Hθ(D))
+of any Gaussian random field with positive definite covariance function g have a representation as
+the solution to the equation
+AZ = W,
+where W is white noise on L2(D) and A = C−1/2 with
+(Cϕ)(x) =
+�
+D
+g(x, y)ϕ(y) dµ(x),
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+3
+see [28, Proposition 2.3] for an explicit derivation. Vice versa, any self-adjoint and positive definite
+operator A: Hθ(D) → L2(D) yields a covariance operator C = A−2 with covariance function g
+given as the Schwartz kernel of C. For example, the well known Mat´ern covariance kernels [40] are
+given through D = Rd and A = (κ2 − ∆)θ/2 with κ > 0, θ > d/2, and are asymptotically smooth.
+More generally, we may consider any self-adjoint and positive definite pseudo-differential operator
+A ∈ OPSθ
+cl,δ(D) of order θ > d/2 with symbol of Gevrey class δ ≥ 1 in the sense of [8, Definition
+1.1]1. This implies C = A−2 ∈ OPS−2θ
+cl,δ (D) as a consequence of the pseudo-differential operator
+calculus for Gevrey classes developed in [8]. In analogy to [47, Lemma 3.0.2] we obtain that the
+covariance kernel g (i.e., the Schwartz kernel) of C is smooth away from the diagonal and satisfies
+|∂α
+x ∂β
+y g(x, y)| ≤ CGA|α|+|β|(α!β!)δ∥x − y∥2θ−d−|α|−|β|
+2
+,
+x, y ∈ D, x ̸= y,
+(1)
+for all α, β ∈ Nd and kernel dependent constants CG, A > 0. We note that the special case δ = 1
+corresponds to the classical asymptotical smoothness. For δ ≥ 1 we will refer to Gδ-asymptotical
+smoothness and call the kernel function a Gevrey kernel.
+These considerations make clear that a unified treatment of asymptotically smooth and more
+generally Gδ-asymptotically smooth covariance functions as presented in this article is desirable.
+1.4. Contributions. The objective of this article is to present an algorithm with rigorous error
+bounds and complexity estimates for estimating Gevrey kernels and covariance functions in op-
+timal complexity. This will be achieved by using a multilevel sample covariance estimator on an
+approximate multilevel hierarchy of H2-matrices. More precisely
+• we generalize the variable-order H2-approximation theory, see [4, 7, 6], to Gδ-asymptotically
+smooth kernels. The basis for this generalization is a new approximation result for Gevrey
+regular functions.
+• we develop a multilevel algorithm which allows to evaluate the sample covariance estima-
+tor in variable-order H2-compressed form with negligible approximation error in optimal
+complexity.
+• we provide numerical examples which estimate covariance matrices with tens of billions of
+entries, underlying the feasibility of the proposed algorithm.
+One of the major implications of these contributions is that Gδ-asymptotically smooth covariance
+functions of a Gaussian processes can now be asymptotically estimated with the same complexity
+as the mean. We also note that variable-order results imply fixed order results as a special case.
+1.5. Outline. The article is organized as follows.
+First, in Section 2, we provide a new ap-
+proximation result for Gevrey-regular functions and use this result for establishing the required
+variable-order H2-approximation rates for Gevrey kernels. These results are then used in Section 3
+for establishing approximation rates of a single-level H2-formatted sample covariance estimator
+and its computational realization. Section 4 is concerned with the construction and analysis of
+the H2-formatted multilevel sample covariance estimator, whereas Section 5 considers its algorith-
+mic implementation. Finally, in Section 6, we provide the numerical experiments underlining our
+theoretical considerations before we draw our conclusions in Section 7.
+2. H2-approximation of Gevrey kernels
+2.1. Interpolation of Gevrey functions. We start our considerations by recalling the definition
+of functions of Gevrey class and some basic facts on polynomial interpolation.
+Definition 2.1. Let D ⊂ Rd and f ∈ C∞(D). f is of Gevrey class δ ≥ 1 with CG, A > 0,
+f ∈ Gδ(D, CG, A), if for every K ⋐ D and α ∈ Nd it holds
+|∂αf(x)| ≤ CGA|α|(α!)δ
+for all x ∈ K.
+A function is analytic, if it is of Gevrey class δ = 1.
+1We refrain from making this notion more explicit as we will not need it for the remainder of the article.
+
+4
+J. D ¨OLZ
+Assumption 2.2. The polynomial interpolation I[a,b]
+m
+: C([a, b]) → Pm on m + 1 distinct points
+in [a, b] is stable, i.e.,
+��I[a,b]
+m
+[f]
+��
+C([a,b]) ≤ Λm∥f∥C([a,b]),
+for all m ∈ N, with stability constant Λm ≥ 1.
+An example satisfying this assumption is the interpolation on Chebychev points, which is stable
+with stability constant Λm ≤ 2
+π ln(m) + 1, see, e.g., [45, Theorem 1.2].
+Lemma 2.3 ([4, Lemma 4.13]). For m ∈ N and f ∈ C([a, b]) it holds
+��f − I[a,b]
+m
+[f]
+��
+C([a,b]) ≤ (Λm + 1) min
+p∈Pm ∥f − p∥C([a,b]).
+The following theorem is the main result of this subsection. In comparison to other approx-
+imation results in the literature, we note that the dependence of the contraction factor on A is
+explicit. This is an essential ingredient for establishing the H2-approximation rates later on.
+Theorem 2.4. Let f ∈ Gδ([−1, 1], CG, A), ρ(r) = r +
+√
+1 + r2, and m ∈ N, m ≥ 3. Then it holds
+min
+p∈Pm ∥f − p∥C([−1,1]) ≤ C(A, δ)CGρ(1/A)−m1/δ/e2,
+where C(A, δ) is monotonically increasing in A.
+Proof. The proof is inspired by the one of [43, Proposition 4.1]. Denote by I3 : H2([−1, 1]) → P3
+the Hermite interpolation operator given by I3f(±1) = f(±1), (I3f)′(±1) = f ′(±1) and, for
+m ∈ N, m ≥ 3, denote by πm−2,0 : L2([−1, 1]) → Pm−2 the L2-orthogonal projection onto the first
+m − 1 Legendre polynomials. Then, the projector H2([−1, 1]) → Pm defined by
+(πm,2f)(x) = (I3f)(x) +
+� x
+−1
+� y
+−1
+�
+πm−2,0
+�
+(f − I3f)′′��
+(z) dz dy
+satisfies the error estimate, see [15, Theorem A.1],
+∥f − πm,2f∥2
+H2([−1,1]) ≤ C (m − 1 − k)!
+(m − 1 + k)!
+��f (k+2)��2
+L2([−1,1]),
+2 ≤ k ≤ m − 1.
+Now, fix α = (2ρ(1/A)ρ(A)A)−1/δ, k = ⌊αγm1/δ⌋ with γ = min{max{
+2
+αm1/δ , 1},
+m−1
+αm1/δ }, and
+note that 2 ≤ k ≤ m − 1, k ≤ αγm1/δ ≤ k + 1, and ρ(1/A)ρ(A) ≤ (2/A + 1)2A+1 =: Ξ(A). Gevrey
+regularity f ∈ Gδ([−1, 1], CG, A) and Stirling’s formula
+√
+2πn(n/e)n ≤ n! ≤ e√n(n/e)n, n ∈ N
+imply
+∥f − πm,2f∥2
+H2([−1,1]) ≤ CC2
+GA2k+4 (m − 1 − k)!
+(m − 1 + k)!
+�
+(k + 2)!
+�2δ
+≤ CC2
+GA2k+4 e1+2k
+√
+2π
+(m − 1 − k)m−1−k+1/2
+(m − 1 + k)m−1+k+1/2
+�
+k!(k + 2)2�2δ
+≤ CC2
+GA2k+4 e1+2k
+√
+2π
+(m − 1 − k)m−1−k+1/2
+(m − 1 + k)m−1+k+1/2 e2δ(1−k)k2kδkδ(k + 2)4δ
+≤ CC2
+GA2k+4 e1+2δ+2(1−δ)k
+√
+2π
+�m − 1 − k
+m − 1 + k
+�m−1−k+1/2
+m−2kk2kδkδ(k + 1)4δ.
+Since 1 − δ ≤ 0, m − 1 − k + 1/2 ≥ 0 for k ≤ m − 1, m−k ≤ (αγ/k)δk, and kδ(k + 2)4δ ≤ C(δ)22k
+for k ≥ 2 this implies
+∥f − πm,2f∥H2([−1,1]) ≤ C(δ)CGAk+2γδkαδk2k.
+We next remark that γδk ≤ 1 for 2 ≤ αm1/δ.
+For for 2 > αm1/δ, we remark that γδk ≤
+γ2δ ≤ C(δ)(Ξ(A)A)2, where Ξ(A)A is continuous and monotonically increasing on (0, ∞) with
+limt→0 Ξ(t)t = 2. Thus, γδk ≤ χ(A, δ) is monotonically increasing in A with χ(A, δ) ≥ 4C(δ).
+The continuous embedding H2([−1, 1]) �→ L∞([−1, 1]) and the definition of α then yield
+∥f − πm,2f∥C([−1,1]) ≤ C(A, δ)CGA2ρ(1/A)ρ(A)ρ(1/A)−ρ(A)(k+1) ≤ C(A, δ)CGρ(1/A)−ρ(A)αγm1/δ,
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+5
+where C(A, δ) is monotonically increasing in A. To obtain the desired exponent, we consider that
+ρ(A)α(A, δ) is monotonically increasing in δ, and that it is bounded from below by e−2 for δ = 1.
+For αm1/δ < m − 1 this yields ρ(A)αγm1/δ ≥ m1/δ/e2 due to γ ≥ 1. For αm1/δ ≥ m − 1 we
+observe that ρ(A)αγm1/δ ≥ m − 1 ≥ m1/δ/e2, which yields the assertion.
+□
+Corollary 2.5. For any f ∈ Gδ([a, b], CG, A), B = A(b − a)/2, and m ∈ N, m ≥ 3, it holds that
+��f − I[a,b]
+m
+[f]
+��
+C([a,b]) ≤ C(B, δ)CG(Λm + 1)ρ(1/B)−m1/δ/e2,
+where C(B, δ) is monotonically increasing in B.
+Proof. Denoting Φ[a,b] : [−1, 1] → [a, b] with Φ[a,b](t) = (b + a)/2 + t(b − a)/2, one easily verifies
+that f ∈ Gδ([a, b], CG, A) implies f ◦ Φ[a,b] ∈ Gδ([−1, 1], CG, B). Lemma 2.3 and Theorem 2.4
+yield the assertion.
+□
+We close the subsection by generalizing the result to tensor product domains in higher dimen-
+sions.
+Definition 2.6. For Q = ×
+d
+i=1[ai, bi], f ∈ C(Q), and m ∈ N, we define the tensor product
+interpolation operator IQ
+m = �d
+i=1 IQ
+m,i, with IQ
+m,i denoting the action of Im in coordinate direction
+i = 1, . . . , d of Q.
+Theorem 2.7. Let Q = ×
+d
+i=1[ai, bi], f ∈ Gδ(Q, CG, A), and m ∈ N, m ≥ 3. Then it holds
+∥f − IQ
+m[f]∥C(Q) ≤ C(A diam∞(Q)/2, δ)CGd(Λm + 1)dρ
+�
+2
+A diam∞(Q)
+�−m1/δ/e2
+,
+where C(A, δ) is monotonically increasing in A.
+Proof. In complete analogy to the proof of [4, Corollary 4.21], using Corollary 2.5 and Assump-
+tion 2.2.
+□
+2.2. Interpolation of Gevrey kernels. As outlined in Section 1.3, it is desirable to generalize
+the approximation theory of the widely known class of asymptotically smooth kernel functions to
+kernels satisfying the following definition.
+Definition 2.8. Let Dx, Dy ⊂ Rd and g ∈ C∞({(x, y) ∈ Dx × Dy : x ̸= y}). For δ ≥ 1, g is
+called Gδ(CG, A)-asymptotically smooth on Dx × Dy if there exist CG, A > 0 and q ∈ R such that
+it holds
+|∂α
+x ∂β
+y g(x, y)| ≤ CGA|α|+|β|(α!β!)δ∥x − y∥−2q−d−|α|−|β|
+2
+,
+x ∈ Dx, y ∈ Dy, x ̸= y,
+(2)
+for all α, β ∈ Nd. For δ = 1 we obtain the classical asymptotical smoothness.
+The following theorem generalizes the very similar result for asymptotically smooth kernels
+proven in [4, Theorem 4.22].
+Theorem 2.9. Let Qt = ×
+d
+i=1[ai, bi] and Qs = ×
+2d
+i=d+1[ai, bi]. Let η > 0 and Qt and Qs be
+admissible, i.e.,
+max
+�
+diam∞ Qt, diam∞ Qs} = diam∞(Qt × Qs) ≤ 2η dist2(Qt, Qs).
+(3)
+Let g be Gδ(CG, A)-asymptotically smooth on Qt × Qs and ˜g = IQt×Qs
+m
+[g]. Then it holds for
+m ∈ N, m ≥ 3,
+∥g − ˜g∥C(Qt×Qs) ≤ C(Aη, δ)CG
+2d(Λm + 1)2d
+dist2(Qt, Qs)2q+d ρ
+� 1
+Aη
+�−m1/δ/e2
+.
+(4)
+Proof. In complete analogy to the proof of [4, Theorem 4.22].
+□
+
+6
+J. D ¨OLZ
+To improve readability we may note that limt→∞ p(t)˜ρt1/δ = 0 for any polynomial p and ˜ρ ∈
+(0, 1) to follow [4, Remark 4.23] and reformulate Equation (4) in Theorem 2.9 as
+∥g − ˜g∥C(Qt×Qs) ≤
+Cin
+dist2(Qt, Qs)2q+d ˜ρm1/δ,
+˜ρ : = min
+�
+Aη
+Aη + 1, Aη
+2
+�1/e2
+> ρ
+� 1
+Aη
+�−1/e2
+,
+(5)
+for some fixed Cin > 0.
+All further results from the classical theory for asymptotically smooth kernels are generalized
+with only minor modifications. In the following subsection we highlight a result going back to [6]
+which allows to choose the polynomial degree of the interpolation according to the spatial size of
+the clusters, yielding linear storage complexity for the compression of Gevrey kernels.
+Remark 2.10. The classical results for asymptotically smooth kernel functions depend on the
+analyticity of the kernel function in admissible clusters since these estimates are based on analytic
+continuations into Bernstein ellipses in the complex plane.
+In contrast, the arguments of our
+generalizations to Gδ(CG, A)-asymptotically smooth kernels only require finite smoothness in each
+direction and do not require extensions into the complex plane.
+2.3. Cluster trees and block-cluster trees. Cluster trees and block-cluster trees are the basis
+for H2-approximations of kernel functions.
+We recall the basic notions along the lines of [27,
+Chapter 5.3, 5.5, and A.2] and [4, Chapter 3.8].
+Definition 2.11. Let I ⊂ N be a finite index set. The cluster tree TI is a tree whose vertices
+correspond to non-empty subsets of I and are referred to as clusters. We require that the root of
+TI corresponds to I and that it holds ˙∪s∈children(t)s = t for all non-leaf clusters t ∈ TI. The leafs
+of TI are denoted by LI and the distance of a cluster t ∈ TI to the root is denoted by level(t) ∈ N.
+The depth of the cluster tree is the maximal level of its clusters.
+Let D ⊂ Rd be bounded and {Di}i∈I a decomposition of D into simply connected sets indexed
+by I. We say that Qt = ×
+d
+i=1[ai, bi] is a bounding box of t if
+Dt = ∪i∈tDi ⊂ Qt,
+for all t ∈ TI.
+We remark that the definition implies that LI provides a decomposition of I. Further, for
+computational reasons, we make the following assumptions on the considered cluster trees.
+Assumption 2.12. Let TI be a cluster tree. We assume that
+(1) the cluster tree is built on a decomposition {Di}i∈I of D ⊂ Rd bounded into simply con-
+nected sets,
+(2) the number of children for non-leaf clusters bounded from below and above, i.e.,
+2 ≤ | children(t)| ≤ Cab,
+t ∈ TI \ LI,
+(6)
+for some Cab > 0,
+(3) the cardinality of the leaf clusters is bounded from below and above, i.e.,
+nmin/Cab ≤ |t| ≤ nmin,
+t ∈ LI,
+(7)
+for some nmin > 0.
+Most standard algorithms for constructing cluster trees result in cluster trees satisfying these
+conditions, see also [4, 27].
+Definition 2.13. Given a cluster tree TI, the block-cluster tree TI×I is a tree with vertices
+corresponding to cluster pairs, referred to as block-clusters.
+Starting with t × s = I × I the
+block-cluster tree is constructed as follows.
+(1) Check whether t × s has admissible bounding boxes in the sense of Equation (3).
+(2)
+(a) If t × s has admissible bounding boxes, add it to L+
+I×I.
+(b) Otherwise, perform Item 1 for all t′ × s′, t′ ∈ children(t), s′ ∈ children(s). If t or s
+have no children, add t × s to L−
+I×I.
+The algorithm induces a tree structure TI×I whose set of leafs is given as LI×I = L+
+I×I ∪ L−
+I×I.
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+7
+0.00
+0.25
+0.50
+0.75
+1.00
+0.0
+0.5
+1.0
+Interpolation
+0.00
+0.25
+0.50
+0.75
+1.00
+0.0
+0.5
+1.0
+Reinterpolation
+0.00
+0.25
+0.50
+0.75
+1.00
+0.0
+0.5
+1.0
+Second reinterpolation
+0.00
+0.25
+0.50
+0.75
+1.00
+0.00
+0.25
+0.50
+0.75
+1.00
+Third reinterpolation
+Figure 1.
+Illustration of iterated interpolation. The continuous polynomial
+(upper left) is replaced by a piecewise polynomial of lower degree (lower right).
+We remark that the definition implies that LI×I provides a partition of I × I. Moreover, if
+t × s ∈ TI×I, then also s × t ∈ TI×I, i.e., the block-cluster tree is symmetric. The following
+constant allows to quantify the sparsity of a block-cluster tree.
+Definition 2.14. Given a block-cluster tree TI×I, its sparsity constant Csp is defined as
+Csp = max
+t∈TI
+���
+s ∈ TI : t × s ∈ TI×I
+���.
+2.4. Variable-order H2-approximation spaces of Gevrey kernels. The following definitions
+aim at defining H2-approximation spaces of kernel functions.
+Definition 2.15. Let TI be a cluster tree and LI its leafs. For all t, s ∈ TI we define
+Lt = {t0 ∈ LI : ∃ cluster chain t0 ⊆ . . . ⊆ tn = t with ti−1 ∈ children(ti), i = 1, . . . , n},
+and
+Lt×s = {t0 × s0 : t0 ∈ Lt, s0 ∈ Ls}.
+Let q ∈ (0, 1).
+The family of bounding boxes (Qt)t∈TI is called q-regular if all cluster chains
+t0 ⊆ . . . ⊆ tn = t, t ∈ TI, t0 ∈ Lt, yield families of bounding boxes (Qi)n
+i=0, Qi = ×
+d
+j=1 Ji
+j
+bounding box to ti, satisfying |Ji−1
+j
+| ≤ q|Ji
+j| for all i = 1, . . . , n, j = 1, . . . , d.
+Definition 2.16. Let TI be a cluster tree and (Qt)t∈TI a q-regular family of bounding boxes. Let
+α ∈ N0, β ∈ N and kδ
+i = ⌈(β + αi)δ⌉. Let t, s ∈ TI, t0 ∈ Lt, s0 ∈ Ls and t0 ⊆ . . . ⊆ tn = t and
+s0 ⊆ . . . ⊆ sm = s cluster chains in TI. We define the interpolation operators
+It
+t0 = It0 ◦ . . . ◦ Itn,
+with Iti = IQi
+kδ
+p−level(ti) for i = 0, . . . , n,
+and
+It×s
+t0×s0 = It
+t0 ⊗ Is
+s0.
+An illustration of the iterated interpolation process can be found in Figure 1.
+Assumption 2.17. We asume that TI is a cluster tree of depth p. In accordance with [4, 6] we
+assume that
+
+8
+J. D ¨OLZ
+(1) there are constants CΛ, λ ≥ 1 such that the stability constant Λm of the interpolation
+operator I[a,b]
+m
+, cf. Assumption 2.2, satisfies Λm ≤ CΛ(m + 1)λ for all m ∈ N0,
+(2) (Qt)t∈TI is a q-regular family.
+Remark 2.18. [4, 6] also assume that TI×I is locally homogeneous. This condition is automati-
+cally satisfied for all block-clusters as constructed in Definition 2.13.
+We are now in the position to define H2-spaces of kernel functions.
+Definition 2.19. Let TI be a cluster tree of depth p with a q-regular family of bounding boxes.
+Let α ∈ N0, β ∈ N, kδ
+i = ⌈(β + αi)δ⌉ and TI×I be a block-cluster tree constructed from TI. We
+define
+Pt×s =
+�
+Pkδ
+p−level(t) ⊗ Pkδ
+p−level(s)
+���
+t×s
+for all t, s ∈ TI,
+Ppw
+t×s = {f : t × s → R: f = It×s
+t0×s0p, t0 × s0 ∈ Lt×s, p ∈ Pt×s}
+for all t × s ∈ L+
+I×I. We define the H2-space of kernel functions as
+V H =
+�
+g: D × D → R: k
+��
+t×s ∈ Ppw
+t×s for all t × s ∈ L+
+I×I
+�
+.
+We remark that the definition implies that each cluster t ∈ TI contains
+Kt =
+�
+kδ
+p−level(t)
+�d =
+�
+(β + α(p − ℓ))δ�d
+(8)
+interpolation points.
+All further results from the variable-order H2-theory for asymptotically smooth kernels are
+generalized with minor modifications. In the following we use the common assumptions and state
+a slightly modified error estimate in the L2-norm, rather than the maximums norm.
+2.5. L2-error of variable-order H2-approximations. For Gevrey-regular kernels, the approx-
+imation error in each block-cluster can be estimated as follows.
+Corollary 2.20. Let Assumption 2.17 hold. Let 2q ∈ [−d, 0), let the kernel function g: Rd×Rd →
+R be Gδ(CG, A)-asymptotically smooth, and let α ∈ N0. Then there are constants Cin ∈ R>0 and
+β0 ∈ N0 such that
+��g − It×s
+t0×s0g
+��
+C(Qt0×Qs0) ≤ Cin
+� ˜ρβ+α(p−level(t))
+diam∞(Qt)2q+d
+�1/2� ˜ρβ+α(p−level(s))
+diam∞(Qs)2q+d
+�1/2
+holds with ˜ρ as in Equation (5) for all β ≥ β0, all blocks t × s ∈ L+
+I×I satisfiyng Equation (3), and
+all t0 ∈ Lt, s0 ∈ Ls.
+Proof. The proof follows the arguments of [6] and [4, Chapter 4.7] with only minor modifications.
+□
+Remark 2.21. The restriction on 2q can be lifted to 2q ∈ R<0, if t × s ∈ L+
+I×I, t ∈ children(t′),
+s ∈ children(s′), t′, s′ ∈ TI, and t′ × s′ does not satisfy Equation (3). This is the case for most
+block-cluster trees, in particular for the ones constructed as in Definition 2.13.
+Although the results from the literature can be generalized to Gevrey kernels, most of the
+analysis in the literature is based on an C(Qt0 × Qs0)-type estimate, which is not compatible with
+the L2-setting of the Monte Carlo type error analysis, for which an L2-estimate is preferable.
+Definition 2.22. Let µ be a measure on D with a suitable σ-algebra. We write L2(D) = L2
+µ(D).
+Moreover, to shorten notation, we assume that D × D is equipped with the product measure ˜µ and
+write L2(s × t) = L2
+˜µ(Ds × Dt) for any t × s ∈ TI×I.
+We remark that the assumptions on D and its measure are quite general, covering manifolds,
+graphs, and multi-screens as well as point measures, for example.
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+9
+Assumption 2.23. In addition to Assumption 2.17 we assume that there are constants Ccu ∈
+R>0, hH ∈ R>0, Cgr ∈ R>0, and ζ ∈ R≥1 such that
+µ(Dt) ≤ Ccu diam∞(Qt)d,
+for all t ∈ TI,
+C−1
+gr hH ≤ diam∞(Qt) ≤ CgrhH
+for all t ∈ LI, and
+diam∞(Qt) ≤ ζ diam∞(Qt′)
+for all t′ ∈ children(t), t ∈ TI, see also [4, (4.58) and (4.59)].
+Corollary 2.24. Let Assumption 2.23, and the assumptions of Corollary 2.20 hold. Then it holds
+����g −
+�
+t0×s0∈Lt×s
+It×s
+t0×s0g
+����
+L2(t×s)
+≤ Clch−2q
+H
+˜ρβ(ζ−2q ˜ρα)p−level(t)/2−level(s)/2,
+where Clc = CinCcuC−2q
+gr
+.
+Proof. Assumption 2.23 implies
+diam∞(Qt) ≤ ζlevel(t′)−level(t) diam∞(Qt′) ≤ CgrhHζp−level(t)
+for all t′ ∈ Lt, t ∈ TI. Thus,
+µ(Dt)
+diam∞(Qt)2q+d ≤
+Ccu
+diam∞(Qt)2q ≤ CcuC−2q
+gr
+h−2q
+H ζ−2q(p−level(t)).
+The assertion follows from H¨olders inequality and Corollary 2.20 due to
+����g −
+�
+t0×s0∈Lt×s
+It×s
+t0×s0g
+����
+L2(t×s)
+≤
+max
+t0×s0∈Lt×s ∥g − It×s
+t0×s0g∥C(Qt0×Qs0)µ(Dt)1/2µ(Ds)1/2
+≤ Cin
+�µ(Dt)˜ρβ+α(p−level(t))
+diam∞(Qt)2q+d
+�1/2�µ(Ds)˜ρβ+α(p−level(s))
+diam∞(Qs)2q+d
+�1/2
+.
+□
+2.6. Storage requirements of H2-farfield approximations. The following estimate on the
+storage requirements of the farfield of variable-order H2-approximations follows.
+Lemma 2.25. Let Assumption 2.12 and Assumption 2.23 hold. Let g ∈ V H with α ∈ N0 and
+β ∈ N. Then the storage requirements for the coefficients of all leafs t ∈ L+
+I×I are bounded by
+CH2((α + β)δd|I|),
+i.e., they are linear with respect to the cardinality of the underlying index set I. The constant
+CH2 is independent of the depth of TI×I and depends only on δ, d, Csp, and the shape of TI (see
+Appendix A for a precise statement).
+Proof. We use the framework provided in [4, Chapter 3.8]. Lemma A.2 shows that the rank as given
+by Equation (8) yields a (1, α, β, δd, Cab)-bounded rank distribution in the sense of [4, Definition
+3.44], see also Definition A.1. Lemma A.4 yields that TI is a (Crc, α, β, δd, Cab)-regular cluster
+tree in the sense of [4, Definition 3.47], see also Definition A.3, with Crc given as in Equation (26).
+The assertion follows from [4, Corollary 3.49], see also Lemma A.7.
+□
+
+10
+J. D ¨OLZ
+3. H2-sample covariance estimation
+3.1. Approximation of Gaussian random field samples. We consider finite dimensional
+approximation spaces Vh ⊂ L2(D), h > 0, and denote the L2-projection onto Vh by Πh : L2(D) →
+Vh. The approximation spaces are assumed to satisfy the approximation estimate
+∥u − Πh∥L2(D) ≤ CL2hγ∥u∥Hγ(D),
+for all u ∈ Hγ(D),
+(9)
+for all 0 ≤ γ ≤ m for some m ∈ N with the Hilbert spaces Hγ(D) ⊂ L2(D) appropriately chosen
+such that Hγ(D) ⊂ Hγ′(D) ⊂ L2(D), 0 ≤ γ′ ≤ γ ≤ m. These approximation estimates hold in
+scattered data approximation [50] and for the standard piecewise polynomial finite element spaces
+of polynomial degree m on quasi uniform meshes on manifolds or graphs [9] with Hm(D) being
+the standard Sobolev spaces, for example.
+Denoting by ⊗ the Hilbertian tensor product, we identify L2(D × D) ≃ L2(D) ⊗ L2(D) and
+write Πmix
+h
+= Πh ⊗ Πh for the L2-projection Πmix
+h
+: L2(D × D) → Vh ⊗ Vh. We further introduce
+the spaces of mixed regularity Hθ
+mix(D × D) = Hθ(D) ⊗ Hθ(D) for θ > 0 and note that for any
+given centered Gaussian random field Z ∈ L2
+P(Ω; Hθ(D)) it holds
+g = E[Z ⊗ Z] ∈ Hθ
+mix(D × D)
+for its covariance function g due to
+∥g∥Hθ
+mix(D×D) =
+��E[Z ⊗ Z]
+��
+Hθ
+mix(D×D) ≤ ∥Z ⊗ Z∥L1
+P(Ω;Hθ
+mix(D×D)) ≤ ∥Z∥2
+L2
+P(Ω;Hθ(D)),
+(10)
+see also [14, Equation (4.10)], for example.
+Lemma 3.1. Let Z ∈ L2
+P(Ω; Hθ(D)), θ > 0, be a Gaussian random field and g ∈ Hθ
+mix(D) its
+covariance function.
+Let Vh be an approximation space such that Equation (9) holds for γ =
+min{θ, m}. Then there is a constant C⊗
+L2 ∈ R>0 depending on CL2 such that it holds
+∥g − Πmix
+h
+g∥L2(D×D) ≤ C⊗
+L2hγ∥g∥Hγ
+mix(D×D) ≤ C⊗
+L2hγ∥Z∥2
+L2
+P(Ω;Hγ(D)).
+Proof. The first estimate is standard, the second follows from Equation (10).
+□
+3.2. L2-projection onto H2-space. Given the discrete approximation in a tensor product ap-
+proximation space Vh ⊗ Vh ⊂ L2(D × D) to a Gδ(CG, A)-asymptotically smooth kernel, we would
+like to convert this approximation into a variable-order H2-approximation of the kernel function.
+This is accomplished by L2-projection into the vector space of H2-approximated kernel functions
+V H from Definition 2.19.
+Definition 3.2. We denote the L2-projection of k ∈ L2(D × D) onto V H by ΠHk.
+Remark 3.3. Due to Assumption 2.12 and Assumption 2.23, computing ΠHk is equivalent to
+computing the L2(t × s) projections ΠH
+t×sk of k|t×s onto Ppw
+t×s and setting
+ΠHk =
+�
+t×s∈L+
+I×I
+ΠH
+t×sk +
+�
+t×s∈L−
+I×I
+k|t×s.
+for k ∈ L2(D × D). We extend ΠH
+t×sk and k|t×s by zero outside of t × s to simplify notation.
+Lemma 3.4. The assumptions of Corollary 2.24 together with Remark 3.3 imply
+��g − ΠHg
+��
+L2(t×s) =
+��g − ΠH
+t×sg
+��
+L2(t×s) ≤ Clch−2q
+H
+˜ρβ(ζ−2q ˜ρα)p−level(t)/2−level(s)/2
+for all blocks t × s ∈ L+
+I×I.
+Proof. Follows immediately from C´ea’s lemma and Corollary 2.24.
+□
+Lemma 3.5. Let the assumptions of Corollary 2.24 hold. Choose α ∈ N such that ζ−2q ˜ρα < 1.
+Then there is β0 ∈ N such that
+��g − ΠHg
+��
+L2(D×D) ≤ ClcCsph−2q
+H
+˜ρβ
+1 − ζ−2q ˜ρα
+for all β ≥ β0 with ˜ρ as in Equation (5).
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+11
+Proof. Due to L2(D × D) ≃ L2(D) ⊗ L2(D) ≃ L2(D; L2(D)) we may write
+��g − ΠHg
+��
+L2(D×D) =
+sup
+u,v∈L2(D)
+u,v̸=0
+�
+D
+�
+D
+�
+g(x, y) − ΠHg(x, y)
+�
+u(x)v(y) dµ(x) dµ(y)
+∥u∥L2(D)∥v∥L2(D)
+.
+Using Lemma 3.4, the Cauchy-Schwartz inequality, and sparsity of TI×I the numerator is estimated
+by
+�
+D
+�
+D
+�
+g(x, y) − ΠHg(x, y)
+�
+u(x)v(y) dµ(x) dµ(y)
+≤
+�
+t×s∈L+
+I×I
+��g − ΠHg
+��
+L2(t×s)∥u∥L2(t)∥v∥L2(s)
+≤ Clch−2q
+H
+˜ρβ
+�
+t×s∈L+
+I×I
+(ζ−2q ˜ρα)(p−level(t))/2∥u∥L2(t)(ζ−2q ˜ρα)(p−level(s))/2∥v∥L2(s)
+≤ ClcCsph−2q
+H
+˜ρβ
+� �
+t∈TI
+(ζ−2q ˜ρα)p−level(t)∥u∥2
+L2(t)
+�1/2� �
+s∈TI
+(ζ−2q ˜ρα)p−level(s)∥v∥2
+L2(t)
+�1/2
+≤ ClcCsph−2q
+H
+˜ρβ
+�
+p
+�
+ℓ=0
+(ζ−2q ˜ρα)p−ℓ
+�
+t∈TI
+level(t)=ℓ
+∥u∥2
+L2(t)
+�1/2�
+p
+�
+ℓ=0
+(ζ−2q ˜ρα)p−ℓ
+�
+s∈TI
+level(s)=ℓ
+∥v∥2
+L2(t)
+�1/2
+.
+Finally, ζ−2q ˜ρα < 1 implies
+p
+�
+ℓ=0
+(ζ−2q ˜ρα)p−ℓ
+�
+t∈TI
+level(t)=ℓ
+∥u∥2
+L2(t) ≤ ∥u∥2
+L2(D)
+p
+�
+ℓ=0
+(ζ−2q ˜ρα)ℓ ≤
+∥u∥2
+L2(D)
+1 − ζ−2q ˜ρα ,
+which yields the assertion.
+□
+Corollary 3.6. Let the assumptions of Corollary 2.24 hold and let Vh be an approximation space
+such that Equation (9) holds for γ = min{θ, m}. Choose α ∈ N such that ζ−2q ˜ρα < 1. Then there
+is β0 ∈ N such that
+��g − ΠHΠmix
+h
+g
+��
+L2(D×D) ≤ ClcCsph−2q
+H
+˜ρβ
+1 − ζ−2q ˜ρα
++ C⊗
+L2hγ∥Z∥2
+L2
+P(Ω;Hγ(D))
+for all β ≥ β0 with ˜ρ as in Equation (5).
+Proof. Follows from stability of the L2-projection,
+��g − ΠHΠmix
+h
+g
+��
+L2(D×D) ≤
+��g − ΠHg
+��
+L2(D×D) +
+��g − Πmix
+h
+g
+��
+L2(D×D),
+Lemma 3.1, and Lemma 3.5.
+□
+In the next subsection we discuss how we can apply ΠH to simple tensors with elements in Vh
+in linear complexity in dim(Vh).
+3.3. Algorithmic realization of ΠH applied to simple tensors. As we will see below, com-
+puting ΠH(zh⊗zh), zh ∈ Vh, efficiently is one of the central operations in the H2-formatted (single-
+and multi-level) estimation of covariance functions and thus deserves some discussion. Remark 3.3
+implies that for any zh ∈ Vh we have
+ΠH(zh ⊗ zh) =
+�
+t×s∈L+
+I×I
+ΠH
+t×s(zh|t ⊗ zh|s) +
+�
+t×s∈L−
+I×I
+zh|t ⊗ zh|s,
+where ΠH
+t×s(zh|t ⊗ zh|s) = upw
+t×s ∈ Ppw
+t×s are the solutions of the local variational problems
+Find upw
+t×s ∈ Ppw
+t×s s.t. (upw
+t×s, ppw
+t×s)L2(t×s) = (zh|t ⊗ zh|s, ppw
+t×s)L2(t×s) for all ppw
+t×s ∈ Ppw
+t×s,
+(11)
+for all t × s ∈ L+
+t×s.
+
+12
+J. D ¨OLZ
+Crucially, Ppw
+t×s inherits the tensor product structure of Pt×s, i.e., it holds
+Ppw
+t×s = Ppw
+t
+⊗ Ppw
+s ,
+for all t × s ∈ L+
+I×I, where
+Ppw
+t
+= {f ∈ L2(t): f = It
+t0p, t0 ∈ Lt, p ∈ Pkp−level(t)
+��
+t},
+for all t ∈ TI. Thus, Equation (11) is equivalent to solving the finite dimensional variational
+problems
+Find upw
+r
+∈ Ppw
+t
+s.t. (upw
+r , ppw
+r )L2(r) = (zh|r, ppw
+r )L2(r) for all ppw
+r
+∈ Ppw
+r ,
+for r ∈ {t, s} and setting upw
+t×s = upw
+t
+⊗ upw
+s . Fixing appropriate nodal bases Ppw
+r
+= span{ψr
+i }m
+i=1
+with m as in Equation (8) this is equivalent to solving the systems of linear equations
+Qrur = qh
+r
+(12)
+with
+Qr =
+�
+(ψr
+i , ψr
+j)L2(r)
+�m
+i,j=1,
+qh
+r =
+�
+(zh|r, ψr
+i )L2(r)
+�m
+i=1,
+ur =
+�
+ψr
+i
+�m
+i=1,
+(13)
+for r ∈ {t, s}. The expression for qh
+r can be further simplified to
+qh
+r = Mrzh
+r,
+where Mr =
+�
+(ψr
+i , φr
+j)L2(r)
+�
+i,j, ψr
+i ∈ Ppw
+r , φr
+j ∈ Vj|r, is the moment matrix on r and zh
+r is the
+coefficient vector of zh|r. We note that Ppw
+t
+= Pt for all t ∈ LI.
+We will now show that, for a given sample zh ∈ Vh, computing ΠH(zh⊗zh) can be accomplished
+in O(dim Vh) complexity. To avoid technicalities, we make the following simplifying assumption,
+which is satisfied if Vh is suitably build on refinements of the decomposition {Di}i∈I, for example.
+Assumption 3.7. We assume that dim(Vh|s) ≤ Cminnmin for all s ∈ LI and some constant
+Cmin > 0.
+Definition 3.8. Let t ∈ TI \LI, t′ ∈ children(t), and Et′ be the matrix representation of Et′ : Pt →
+Pt′ defined by p �→ It′p with respect to the bases {ψt
+i}m
+i=1 and {ψt′
+i }m
+i=1. We refer to {Et}t∈TI\{I}
+as the transfer matrices. For the constant order case, i.e., for α = 0, we denote the family of
+transfer matrices by {Ft}t∈TI\{I}.
+Lemma 3.9. Let Assumption 2.12 and Assumption 3.7 hold and let zh ∈ Vh.
+Then we can
+compute {qh
+t }t∈TI defined as in Equation (13) in at most CH2(α + β)δd|I| operations with the
+H2-forward transformation, see, e.g., [4], i.e, as follows:
+(1) Compute qh
+t = Mtzh
+t for all t ∈ LI.
+(2) Recursively compute qh
+t = �
+t′∈children(t) E⊺
+t′qh
+t′ for all t ∈ TI \ LI.
+Proof. This is a classical result from the literature, see [4, Lemma 3.45 and 3.48], using the same
+constants as in the proof of Lemma 2.25.
+□
+Lemma 3.10. Let Assumption 2.12 and Assumption 3.7 hold. We can compute {Qt}t∈TI as
+defined in Equation (13) in in at most 2CH2(α + β)2δd|I| operations as follows:
+(1) Compute Qt for all t ∈ LI. Keep in mind that Ppw
+t
+= Pt in this case.
+(2) Recursively compute Qt = �
+t′∈children(t) E⊺
+t′Qt′Et′ for all t ∈ TI \ LI.
+Proof. In complete analogy to Lemma 3.9, see also Lemma 2.25 and [4, Lemma 3.45 and 3.48].
+□
+We remark that actual implementations would compute and factorize {Qt}t∈TI once and use it
+for all samples, whereas {qt}t∈TI needs to be recomputed for each sample. However, we will not
+further exploit this fact in the following estimates.
+Theorem 3.11. Let Assumption 2.12 and Assumption 3.7 hold and let zh ∈ Vh and TI be a cluster
+tree. Then we can compute ΠH(zh ⊗ zh) in at most 7CH2(α + β)2δd|I| operations as follows:
+(1) Compute {qh
+t }t∈TI and {Qt}t∈TI as in Lemma 3.9 and Lemma 3.10.
+(2) Solve the local systems Qtut = qh
+t , see Equation (12), for all t ∈ LI.
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+13
+(3) Compute ut ⊗ us to obtain ΠH
+t×s(zh|t ⊗ zh|s) = upw
+t×s ∈ Ppw
+t×s for all t × s ∈ L+
+I×I and
+zh|t ⊗ zh|s for all t × s ∈ L−
+I×I.
+Proof. Computing {qh
+t }t∈TI and {Qt}t∈TI is achivable in a combined 3CH2(α + β)2δd|I|, see
+Lemma 3.9 and Lemma 3.10.
+Solving the local systems t ∈ TI is achievable in at most 3K3
+t
+complexity if a dense solver is used, with Kt given as in Equation (8). [4, Lemma 3.45 and 3.48]
+with the same constants as in the proof of Lemma 2.25 yields that solving all local systems requires
+3CH2(α+β)2δd|I| operations in total. Computing ut ⊗us, t×s ∈ L+
+I×I, requires KtKs operations.
+[4, Lemma 3.49] yields that the third step can be achieved in CH2(α + β)2δd|I| operations. This
+yields the assertion.
+□
+3.4. H2-sample covariance estimation. Consider a centered Gaussian random field Z ∈ L2
+P(Ω; Hθ(D)),
+θ > 0, with unknown covariance function g ∈ Gδ(CG, A). We would like to estimate g in H2-
+compressed form from approximations of i.i.d. samples of Z.
+Definition 3.12. Given an approximation space Vh ⊂ L2(D) we define the sample covariance
+estimator (SCE) as
+E[Πmix
+h
+g] ≈ EMC[Πmix
+h
+g] = 1
+M
+M
+�
+k=1
+Πmix
+h
+�
+z(k) ⊗ z(k)�
+= 1
+M
+M
+�
+k=1
+�
+Πhz(k) ⊗ Πhz(k)�
+,
+with i.i.d. samples z(k), k = 1, . . . , M, M ∈ N, of Z ∈ L2
+P(Ω, Hθ(D)).
+Lemma 3.13. Let Z ∈ L2
+P(Ω; Hθ(D)), θ > 0, be a centered Gaussian random field with covariance
+function g. Let Vh be an approximation space such that Equation (9) holds for γ = min{θ, m}.
+Then it holds
+��g − EMC[Πmix
+h
+g]
+��
+L2
+P(Ω;L2(D×D)) ≤
+�
+C⊗
+L2hγ +
+1
+√
+M
+�
+∥Z∥2
+L2
+P(Ω;Hγ(D)).
+Proof. The estimate is derived by standard methods using Lemma 3.1, see, e.g., also [1].
+□
+As is meanwhile well known, see e.g. [1] for a reference, the naive sample covariance estimator
+from Definition 3.12 is computationally inconvenient for the estimation of second moments since it
+yields a quadratic complexity in the dimension of Vh. Instead, we pursue the following alternative.
+Definition 3.14. The H2-formatted sample covariance estimator (H2-SCE) is defined as
+E[ΠHΠmix
+h
+g] ≈ EMC[ΠHΠmix
+h
+g] = 1
+M
+M
+�
+k=1
+ΠH�
+Πhz(k) ⊗ Πhz(k)�
+.
+As outlined in the previous subsection, a single sample of the estimator can be computed in
+linear complexity in |I| ∼ dim(Vh), if a solver with linear complexity for evaluating Πhz(k) is used.
+Thus, the overall complexity of the H2-SCE is O(M|I|).
+Lemma 3.15. Let the assumptions of Lemma 3.4 and Lemma 3.13 hold. Choose α ∈ N such that
+ζ−2q ˜ρα < 1. Then there is β0 ∈ N such that
+��g − EMC[ΠHΠmix
+h
+g]
+��
+L2
+P(Ω,L2(D×D)) ≤ ClcCsph−2q
+H
+˜ρβ
+1 − ζ−2q ˜ρα
++
+�
+C⊗
+L2hγ +
+1
+√
+M
+�
+∥Z∥2
+L2
+P(Ω;Hγ(D))
+for all β ≥ β0 with ˜ρ as in Equation (5).
+Proof. We first note that EMC[ΠHΠmix
+h
+g] = ΠHEMC[Πmix
+h
+g]. Stability of the L2-projection yields
+��g − EMC[ΠHΠmix
+h
+g]
+��
+L2
+P(Ω,L2(D×D))
+=
+��g − ΠHEMC[Πmix
+h
+g]
+��
+L2
+P(Ω,L2(D×D))
+≤
+��g − ΠHg
+��
+L2(D×D) +
+��g − EMC[Πmix
+h
+g]
+��
+L2
+P(Ω,L2(D×D)).
+The first term is estimated with Lemma 3.5 and the second with Lemma 3.13.
+□
+
+14
+J. D ¨OLZ
+3.5. Computational H2-sample covariance estimation. For computational covariance esti-
+mation one often aims at a discretization of the covariance function rather than the covariance
+itself. In the following we provide error estimates for bilinear forms of type
+a(uh, vh) =
+�
+D
+�
+D
+g(x, y)uh(x)vh(y) dµ(x) dµ(y)
+(14)
+for uh, vh ∈ Wh with Wh ⊂ L2(D) being some approximation space. The canonical applications are
+bilinear forms of Galerkin schemes and Nystr¨om discretizations in scattered data approximation.
+For the latter we chose the approximation space to be a set of dirac distributions on points xi ∈ D,
+i = 1, . . . , N, such that Equation (14) reads
+a(u, v) =
+N
+�
+i,j=1
+g(xi, xj)uivj
+(15)
+for u = [ui]N
+i=1, v = [vi]N
+i=1 ∈ RN, see also [26]. We first provide the error estimate and thereafter
+some assumptions one will usually make on the approximation space Wh in order to achieve linear
+complexity.
+Corollary 3.16. Let the assumptions of Lemma 3.15 hold and let Wh ⊂ L2(D) be an approxi-
+mation space satisfying Equation (9). Choose α ∈ N such that ζ−2q ˜ρα < 1. Then there is β0 ∈ N
+such that����
+�
+D
+�
+D
+�
+g(x, y) − EMC[ΠHΠmix
+h
+g(x, y)]
+�
+uh(x)vh(y) dµ(x) dµ(y)
+����
+L2
+P(Ω)
+≤
+�ClcCsph−2q
+H
+˜ρβ
+1 − ζ−2q ˜ρα
++
+�
+C⊗
+L2hγ +
+1
+√
+M
+�
+∥Z∥2
+L2
+P(Ω;Hγ(D))
+�
+∥uh∥L2(D)∥vh∥L2(D),
+for all uh, vh ∈ Wh and β ≥ β0 with ˜ρ as in Equation (5).
+Proof. The assertion follows from Lemma 3.15 and the Cauchy-Schwarz inequality in L2(D).
+□
+For computational reasons, the basis of the approximation space Wh needs to be local.
+Assumption 3.17. Let Wh = span{φi}i∈I be an approximation space and TI a cluster tree
+constructed on I. We require that all basis functions φi, i ∈ t with t ∈ LI, are supported on Dt,
+but not on Ds for s ̸= t.
+We readily check that the assumption is fulfilled for piecewise constant finite elements on the
+decomposition {Dt}t∈TI and refinements thereof and for Nystr¨om discretizations.
+Definition 3.18. Let Wh = span{φi}i∈I be an approximation space satisfying Assumption 3.17.
+We call A = [a(φj, φi)]i,j∈I with A as in Equation (14) an H2-matrix, if g ∈ V H and A is stored
+in compressed form.
+In complete analogy to Lemma 2.25 and in accordance with the literature we obtain linear
+storage requirements for A.
+Corollary 3.19. Under the assumptions of Corollary 3.16 and Assumption 3.17, the matrix A
+can be stored with a storage requirement of CH2(α + β)δd|I|, i.e., linear in the cardinality of I.
+This yields the following optimal result complexity-result for the H2-SCE.
+Theorem 3.20. Under the assumptions of Theorem 3.11 and Assumption 3.17 the H2-SCE is
+computable in complexity CH2M(α+β)δd|I|, if the H2-matrix addition is used for the summation.
+Proof. Follows from Definition 3.14, Theorem 3.11, and the linear complexity of the H2-matrix
+addition, see [4, Chapter 7.3].
+□
+We remark that methods relying on a sparse grid approximation of the covariance yield a
+complexity which is only linear up to a logarithmic factor, see, e.g., [1].
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+15
+4. Multilevel H2-sample covariance estimation: Construction and error analysis
+4.1. Multilevel hierarchy and cluster trees. To further improve the computational complex-
+ity of the H2-SCE we pursue in the following a multilevel approach.
+Our considerations are
+guided by the characteristics of nested finite element spaces, but can be transferred to other ap-
+proximation spaces providing a suitable multilevel hierarchy. To that end, we note that on a given
+decomposition on D we can always define a finite element space and, by employing an appropriate
+clustering algorithm, a cluster tree such that the following assumption is true.
+Assumption 4.1. Let Vh0 ⊂ L2(D) be a piecewise polynomial finite element space generated from
+the decomposition Th0 = {D(0)
+i
+}i∈I0 and let TI0 be a cluster tree constructed on I0 which satisfies
+Assumption 2.12.
+Under these circumstances we can generate a sequence of nested decompositions {Thℓ = {D(ℓ)
+i }i∈Iℓ}∞
+ℓ=0
+with
+|Iℓ| = |I0|Cℓ
+uni
+(16)
+for some Cuni > 1 and corresponding finite element spaces Vh0 ⊂ Vh1 ⊂ Vh2 ⊂ . . . ⊂ L2(D) in
+the usual way using uniform refinement. We can also construct nested cluster trees {TIℓ}∞
+ℓ=0 by
+repeated uniform refinement of Th0 as follows.
+Definition 4.2. Let Th0 = {D(0)
+i
+}i∈I0 and let TI0 and Th0 satisfy Assumption 4.1. Let {Thℓ =
+{D(ℓ)
+i }i∈Iℓ}∞
+ℓ=0 be a sequence of nested decompositions generated by uniform refinement of Th0.
+Given a cluster tree TIℓ on Iℓ, we define a cluster tree TIℓ+1 on Iℓ+1 as follows:
+• The vertices of TIℓ+1 \ LIℓ+1 are defined by the one-to-one correspondence of the supports
+of the clusters, i.e.,
+t(ℓ+1) ∈ TIℓ+1 \ LIℓ+1 ⇔ there is t(ℓ) ∈ TIℓ such that D(ℓ+1)
+t(ℓ+1) = D(ℓ)
+t(ℓ),
+(17)
+with D(k)
+t
+= ∪i∈tD(k)
+i
+, k = ℓ, ℓ + 1. The tree hierarchy between the vertices of TIℓ+1 \ LIℓ+1
+is naturally given by the tree structure induced by the nestedness of the cluster supports.
+• For all s ∈ LIℓ let ts ∈ TIℓ+1 \ IIℓ+1 be the corresponding cluster satisfying Equation (17)
+and let Tts be a cluster tree on ts satisfying Assumption 2.12 constructed by a cluster-
+ing algorithm with fixed constant C′
+ab in Equation (6). We define the children of ts as
+children(ts) = Lts, implying that
+LIℓ+1 =
+�
+s∈LIℓ
+Lts.
+Definition 4.3. We say that a sequence of cluster trees is nested if Equation (17) holds for all
+ℓ ∈ N0. To simplify notation we write t = t(ℓ) = t(ℓ+1) whenever Equation (17) is satisfied.
+An illustration to Definition 4.2 and Definition 4.3 is given in Figure 2.
+Lemma 4.4. Let the assumptions from Assumption 4.1 hold. Then the sequence of cluster trees
+{Thℓ = {D(ℓ)
+i }i∈Iℓ}∞
+ℓ=0 as defined in Definition 4.2 is nested and satisfies Assumption 2.12 with
+uniform constants for all ℓ ∈ N0.
+Proof. The nestedness of the cluster trees follows by construction. Further, Definition 4.2 implies
+nmin/C′
+ab ≤ |t| ≤ nmin for all t ∈ Lts due to Equation (7). Since Equation (7) also implies that
+|ts| ≤ 4nmin, each cluster tree Tts has at most
+4nmin
+nmin/Cab′ = 4Cab′
+leafs. Thus, TIℓ+1 satisfies Equation (6) with C′′
+ab = max{Cab, 4C′
+ab}.
+□
+The nestedness of the generated cluster trees directly implies that also the the sequence of
+block-cluster trees {TIℓ×Iℓ}∞
+ℓ=1 constructed as in Definition 2.13 is nested. Moreover the leaves of
+the generated block-cluster trees provide a nested sequence of decompositions of I × I and D × D.
+
+16
+J. D ¨OLZ
+I0 = {1, 2, 3}
+{1}
+{2, 3}
+{2}
+{3}
+I1 = {1, . . . , 9}
+{1, 2, 3}
+{1} {2} {3}
+{4, . . . , 9}
+{4, 5, 6}
+{4} {5} {6}
+{7, 8, 9}
+{7} {8} {9}
+D(0)
+1
+D(0)
+2
+D(0)
+3
+D(1)
+1
+D(1)
+2
+D(1)
+3
+D(1)
+4
+D(1)
+5
+D(1)
+6
+D(1)
+7
+D(1)
+8
+D(1)
+9
+Figure 2. Illustration of nested cluster trees TI0 (upper left) and TI1 (upper
+right) in the sense of Definition 4.2 to nested decompositions {D(0)
+i
+}i∈I0 (bottom
+left) and {D(1)
+i
+}i∈I1 (bottom right).
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+25
+25
+25
+25
+25
+25
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+49
+49
+49
+49
+49
+49
+Figure 3. Illustration of three H2-approximation spaces on D × D = [0, 1]2
+for three binary, nested, and perfectly balanced cluster trees. No approximation
+is performed within the red blocks. The blue blocks are approximated by ten-
+sorized iterated interpolation with the inscribed polynomial degree. β = 3, α = 2,
+and δ = 1 were used as parameters in Equation (8) for this example. The H2-
+approximation spaces are not nested, but have a similar structure which leads to
+an approximate multi-level hierarchy.
+The following definition identifies clusters and block clusters which are equivalent in the sense
+that they correspond to the same parts of D and D × D.
+Definition 4.5. To simplify notation we write
+t ∈ TIℓ
+for all
+t ∈ TIℓ+1,
+t × s ∈ TIℓ×Iℓ
+for all
+t × s ∈ TIℓ+1×Iℓ+1,
+and vice versa, whenever the involved clusters satisfy Equation (17).
+We further note that the farfields and the nearfields of nested block-cluster trees do not provide
+nested decompositions of D × D, since only
+t × s ∈ L+
+Iℓ×Iℓ ⇒ t × s ∈ L+
+Iℓ+1×Iℓ+1
+is guaranteed from the construction, see also Definition 2.13 and Definition 4.2. Thus, the sequence
+{V Hℓ}∞
+ℓ=0 of H2-spaces from Definition 2.19 generated by the sequence of block-cluster trees is
+not nested, see also Figure 3 for an illustration. This holds also for the polynomials in the farfield,
+which depend on the depth of the specific block-cluster tree, see also Equation (8), which in turn
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+17
+depends on ℓ. For clarification we write Ppw,ℓ
+t
+= Ppw
+t
+and Ppw,ℓ
+t×s = Ppw
+t×s for the polynomial spaces
+from Definition 2.19 whenever they are constructed from the cluster tree TIℓ.
+As a last remark of this subsection, we use the introduced notation to localize the multilevel
+hierarchy in the finite element spaces by means of the nestedness of the cluster trees.
+Definition 4.6. Let {Vhℓ}∞
+ℓ=0 and {TIℓ}∞
+ℓ=0 be sequences of nested finite element spaces and nested
+cluster trees as in Definition 4.2. Let Jℓ : Vhℓ → Vhℓ+1 be the canonical prolongation operator
+between nested finite element spaces. For t ∈ LIℓ+1 we write Jt for the matrix representation of
+Jℓ|t : Vhℓ|t → Vhℓ+1|t.
+4.2. Multilevel H2-sample covariance estimation. With a suitable (approximate) multilevel
+structure at hand, we now introduce a multilevel version of the H2-SCE. To shorten notation we
+introduce the operator
+ΠH
+h,ℓ = ΠHℓΠmix
+hℓ .
+Definition 4.7. Given the above sequence of finite element spaces and H2-spaces and setting
+Πmix
+h−1g = 0, we define the H2-formatted multilevel sample covariance estimator (H2-MLSCE)
+recursively as
+E[ΠH
+h,Lg] ≈ EML
+L
+[ΠH
+h,Lg] =
+L
+�
+ℓ=0
+ΠHLEℓ
+��
+ΠH
+h,ℓ − ΠH
+h,ℓ−1
+�
+g
+�
+(18)
+with the single level estimators
+Eℓ
+��
+ΠH
+h,ℓ − ΠH
+h,ℓ−1
+�
+g
+�
+=
+1
+Mℓ
+Mℓ
+�
+k=1
+�
+ΠH
+h,ℓ − ΠH
+h,ℓ−1
+��
+z(k) ⊗ z(k)�
+,
+ℓ = 0, . . . , L,
+given by i.i.d. samples z(k), k = 1, . . . , Mℓ, Mℓ ∈ N, of Z ∈ L2
+P(Ω, Hθ(D)).
+Theorem 4.8. Let Z ∈ L2
+P(Ω; Hθ(D)), θ > 0, be a centered Gaussian random field with co-
+variance function g.
+Let Th0 = {D(0)
+i
+}i∈I0 and let TI0 and Th0 satisfy Assumption 4.1.
+Let
+{Thℓ = {D(ℓ)
+i }i∈Iℓ}L
+ℓ=0 and {TIℓ}L
+ℓ=0 be sequences of decompositions with corresponding cluster
+trees as constructed in Definition 4.2 and {Vhℓ}L
+ℓ=0 a nested sequence of piecewise polynomial
+ansatz spaces of order m ∈ N on {Thℓ}L
+ℓ=0. Define γ = min{θ, m} and choose α ∈ N such that
+ζ−2q ˜ρα < 1. Then there is β0 ∈ N such that it holds
+��g − EML
+L
+[ΠH
+h,Lg]
+��
+L2
+P(Ω;L2(D×D)) ≤
+ClcCsp˜ρβ
+1 − ζ−2q ˜ρα
+�
+h−2q
+H,L + (1 + 2−2q)
+L
+�
+ℓ=0
+h−2q
+H,ℓ
+√Mℓ
+�
++ C⊗
+L2
+�
+hγ
+L + (1 + 2γ)
+L
+�
+ℓ=0
+hγ
+ℓ
+√Mℓ
+�
+∥Z∥2
+L2
+P(Ω;Hγ(D))
+for all β ≥ β0 with ˜ρ as in Equation (5).
+Proof. The estimate is proved in the usual way, using Corollary 3.6, see, e.g., also [1], and using
+stability of the L2-projection on the way.
+□
+Corollary 4.9. Let the assumptions of Theorem 4.8 hold, let
+˜γ = min{−2q, γ} = min{−2q, θ, m},
+and choose α ∈ N such that ζ−2q ˜ρα < 1. Then there is β0 ∈ N and a constant
+0 < CMLE = CMLE
+�
+ClcCsp˜ρβ0, ζ−2q ˜ρα, C⊗
+L2, ChH, −2q, γ, ∥Z∥L2
+P(Ω;Hγ(D))
+�
+such that
+��g − EML
+L
+[ΠH
+h,Lg]
+��
+L2
+P(Ω;L2(D×D)) ≤ CMLE
+�
+h˜γ
+L +
+L
+�
+ℓ=0
+h˜γ
+ℓ
+√Mℓ
+�
+for all β ≥ β0 with ˜ρ as in Equation (5).
+
+18
+J. D ¨OLZ
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+25
+25
+25
+25
+25
+25
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+9
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+25
+49
+49
+49
+49
+49
+49
+Figure
+4.
+Illustration of the multilevel reduction algorithm for H2-
+approximation spaces on three different levels. The farfield is projected directly
+onto the finest level, whereas the nearfield is prolongated recursively.
+Proof. The specific construction of {Thℓ}L
+ℓ=0, {TIℓ}L
+ℓ=0, and {Vhℓ}L
+ℓ=0 using uniform refinement
+implies that C−1
+hHhℓ ≤ hH,ℓ ≤ ChHhℓ for ℓ = 0, . . . , L. This yields the assertion.
+□
+In analogy to Corollary 3.16 we obtain the following bound on the bilinear form induced by
+the covariance function. We recall that this also holds for bilinear forms of Nystr¨om type Equa-
+tion (15), if the corresponding assumptions are made.
+Corollary 4.10. Under the assumptions of Corollary 4.9 there is β0 ∈ N such that
+����
+�
+D
+�
+D
+�
+g(x, y) − ΠHLEML
+L
+[Πmix
+hL g(x, y)]
+�
+uh(x)vh(y) dµ(x) dµ(y)
+����
+L2
+P(Ω)
+≤ CMLE
+�
+h˜γ
+L +
+L
+�
+ℓ=0
+h˜γ
+ℓ
+√Mℓ
+�
+∥uh∥L2(D)∥vh∥L2(D),
+for all β ≥ β0 with ˜ρ as in Equation (5).
+5. Multilevel H2-sample covariance estimation: Algorithmic considerations
+In view of a computational implementation of the multilevel H2-MLSCE in Equation (18) we
+require an efficient way to combine the H2-approximations on different levels, i.e., an efficient
+implementation of the sum over the different levels. Reformulating this task, we seek an efficient
+implementation of the multilevel reduction
+ΠHL :
+�
+L×
+ℓ=0
+W H
+ℓ
+�
+→ W H
+L ,
+�
+gHℓ�L
+ℓ=0 �→ ˜gHL = ΠHL
+L
+�
+ℓ=0
+gHℓ,
+(19)
+with
+W H
+ℓ
+=
+�
+ΠHℓvhl : vhl ∈ Vhℓ ⊗ Vhℓ
+�
+,
+ℓ = 0, 1, . . . , L.
+In the following, we will pursue a strategy which is illustrated in Figure 4. To that end, we exploit
+Remark 3.3, i.e. that ΠHL can be represented as a sum of local L2-projections on t × s ∈ LIL×IL.
+It is clear that there is nothing to do if a target block-cluster of ΠHL is inadmissible, i.e., if
+t × s ∈ L−
+IL×IL. If t × s is admissible, i.e., if t × s ∈ L+
+IL×IL, we observe that
+ΠHL
+t×s
+L
+�
+ℓ=0
+gHℓ =
+L
+�
+ℓ=0
+ΠHL
+t×sgHℓ.
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+19
+Thus, we can compute ΠHL
+t×sgHℓ whenever t × s ∈ L+
+IL×IL and t × s ∈ LIℓ×Iℓ. Otherwise, i.e., if
+t×s ∈ L−
+Iℓ×Iℓ and t×s /∈ LIL×IL, we split s×t into far- and nearfield according to the partitioning
+of TIℓ+1×Iℓ+1, project the resulting farfield blocks to level L and add the nearfield blocks to the
+nearfield of level ℓ + 1.
+5.1. Projecting admissible block-clusters to admissible block-clusters. To that end, we
+consider the case where t × s ∈ L+
+Iℓ×Iℓ and t × s ∈ L+
+IL×IL, i.e., t × s is an admissible block-cluster
+in both block-cluster trees.
+For these block-clusters, computing ΠHL|t×sgHℓ|t×s, gHℓ ∈ W H
+ℓ ,
+amounts to the solution of
+Find gHL|t×s ∈ Ppw,L
+t×s
+s.t.
+(gHL|t×s, ppw,L
+t×s )L2(t×s) = (gHℓ|t×s, ppw,L
+t×s )L2(t×s)
+for all ppw,L
+t×s
+∈ Ppw,L
+t×s .
+This is a finite dimensional variational problem which can be written as
+Qsu(L)
+s×tQ⊺
+t = R(L,ℓ)
+s
+u(ℓ)
+s×t
+�
+R(L,ℓ)
+t
+�⊺
+,
+(20)
+with Qr, r ∈ {s, t}, as in Equation (13), u(L)
+s×t and u(ℓ)
+s×t the coefficient matrices of gHL|t×s and
+gHℓ|t×s, and
+R(L,ℓ)
+r
+=
+��
+ψ(r,L)
+i
+, ψ(r,ℓ)
+j
+�
+L2(r)
+�
+i=1,...,K(L)
+r
+,
+j=1,...,K(ℓ)
+r
+∈ RK(L)
+r
+×K(ℓ)
+r ,
+for all ψ(r,L)
+i
+∈ Ppw,L
+r
+and ψ(r,ℓ)
+i
+∈ Ppw,ℓ
+r
+, r ∈ {s, t}.
+5.2. Projecting inadmissible leaf block-clusters to admissible block-clusters. We con-
+sider the case t × s ∈ L−
+Iℓ×Iℓ and t × s ∈ L+
+IL×IL. Upon noting that it holds gHℓ|t×s ∈ Vhℓ|s ⊗ Vhℓ|t
+for all gHℓ ∈ W H
+ℓ
+we readily remark that
+Find gHL|t×s ∈ Ppw,L
+t×s
+s.t.
+(gHL|t×s, ppw,L
+t×s )L2(t×s) = (gHℓ|t×s, ppw,L
+t×s )L2(t×s)
+for all ppw,L
+t×s
+∈ Ppw,L
+t×s ,
+is a finite dimensional variational problem which can be rewritten as
+Qsu(L)
+s×tQ⊺
+t = N(L,ℓ)
+s
+g(ℓ)
+s×t
+�
+N(L,ℓ)
+t
+�⊺
+.
+(21)
+As in the previous subsection, u(L)
+s×t and u(ℓ)
+s×t are the coefficient matrices of gHL|t×s and gHℓ|t×s,
+and
+N(L,ℓ)
+r
+=
+��
+ψ(r,L)
+i
+, φ(r,ℓ)
+j
+�
+L2(r)
+�
+i=1,...,K(L)
+r
+,
+j=1,...,dim(Vhℓ|r)
+∈ RK(L)
+r
+×dim(Vhℓ|r),
+for all ψ(r,L)
+i
+∈ Ppw,L
+r
+and φ(r,ℓ)
+i
+∈ Vhℓ|r, r ∈ {s, t}.
+5.3. Preliminary computational considerations. In view of an efficient solution of Equa-
+tion (20) and Equation (21), an efficient assembly of the matrices R(L,ℓ)
+t
+and N(L,ℓ)
+t
+is mandatory.
+Before we state our algorithm for the multilevel reduction, we would like to make some preliminary
+remarks on how these matrices can be obtained efficiently.
+Lemma 5.1. Let Assumption 2.12, Assumption 3.7 and Assumption 4.1 hold and consider fam-
+ilies of finite element spaces and cluster trees as in Definition 4.2. Compute {Rt}t∈TL with
+(1) Rt = Qt for all t ∈ LIL,
+(2) Rt = �
+t′∈children(t) E⊺
+t′,LRt′Ft′ for all t ∈ TIℓ \ LIℓ,
+and {Nt}t∈TL with
+(1) Nt = Mt for all t ∈ LIL,
+(2) Nt = �
+t′∈children(t) E⊺
+t′,LNt′J⊺
+t′ for all t ∈ TIℓ \ LIℓ.
+
+20
+J. D ¨OLZ
+Then {Rt}t∈TL can be computed in at most 2CH2(α + β)2δd|IL| operations and {Nt}t∈TL can be
+computed in at most 2CH2C2
+minn2
+min(α + β)2δd|IL| operations.
+Proof. Estimating the effort for {Rt}t∈TL is complete analogy to Lemma 3.10. To estimate the
+one for {Nt}t∈TL, we note that the computational effort in each cluster t′ ∈ children(t) is bounded
+by
+Cminnmin
+�
+K(L)
+t
+K(L)
+t′
++ CminnminK(L)
+t′
+�
+≤ 2C2
+minn2
+minK(L)
+t
+K(L)
+t′ .
+The effort is then bounded in analogy to the one of {Rt}t∈TL.
+□
+The following lemma extends these considerations to the case when an multilevel hierarchy of
+H2-approximation spaces is used.
+Lemma 5.2. Given {Rt}t∈TL and {Nt}t∈TL as in Lemma 5.1 and 0 ≤ ℓ ≤ L, compute {R(L,ℓ)
+t
+}t∈TIℓ
+by
+(1) R(L,ℓ)
+t
+= Rt for all t ∈ LIℓ,
+(2) R(L,ℓ)
+t
+= �
+t′∈children(t) E⊺
+t′,LR(L,ℓ)
+t′
+Et′,ℓ for all t ∈ TIℓ \ LIℓ,
+and {N(L,ℓ)
+t
+}t∈TIℓ by
+(1) N(L,ℓ)
+t
+= Nt for all t ∈ LIℓ,
+(2) N(L,ℓ)
+t
+= �
+t′∈children(t) E⊺
+t′,LN(L,ℓ)
+t′
+Jt′ for all t ∈ TIℓ \ LIℓ.
+Then {R(L,ℓ)
+t
+}t∈TIℓ can be computed in at most
+2CH2 (α(L − ℓ + 1) + β)3δd
+(α + β)δd
+|Iℓ|.
+operations and {N(L,ℓ)
+t
+}t∈TIℓ can be computed in at most
+2CH2C2
+minn2
+min
+(α(L − ℓ + 1) + β)3δd
+(α + β)δd
+|Iℓ|.
+operations.
+Proof. We first note that TIℓ is a (Crc, α, β+(L−ℓ)α, δd, Cab)-bounded as well as a (Crc, α, β, δd, Cab)-
+regular cluster tree with Crc as in Equation (26). Lemma A.7 yields the assertion for {R(L,ℓ)
+t
+}t∈Tℓ.
+Modifying the proof of Lemma 5.1 with similar arguments yields the assertion for {N(L,ℓ)
+t
+}t∈TIℓ .
+□
+5.4. The multilevel H2-reduction algorithm.
+Theorem 5.3. Let Cab be the uniform constant satisfying Equation (6) for all elements of the
+family of cluster trees {TIℓ}L
+ℓ=0 constructed in the proof of Lemma 4.4. Then there is a constant
+CML = CML(CH2, Cmin, Cab, Cuni, nmin, δ, d) such that the computational cost of Equation (19) are
+bounded by
+CML
+(α + β)⌈3δd⌉
+(α + β)δd |IL|,
+i.e., in linear complexity w.r.t. |IL|, if Equation (19) is computed as follows:
+(1) Set ˜gHL = gHL
+(2) Initialize {Qt}t∈TIL , {Rt}t∈TIL , and {Nt}t∈TIL as in Lemma 3.10 and Lemma 5.1
+(3) For ℓ = 0, . . . , L − 1 proceed as follows:
+(a) Initialize {R(L,ℓ)
+t
+}t∈TIℓ and {N(L,ℓ)
+t
+}t∈TIℓ as in Lemma 5.2
+(b) Project all far- and nearfield blocks on level ℓ to level L, i.e., set
+˜gHL|t×s = ˜gHL|t×s + ΠHL|t×sgHℓ|t×s
+for all t × s ∈ LIℓ×Iℓ with t × s ∈ LIL×IL, by solving the local systems Equation (20)
+and Equation (21).
+(c) For all t × s ∈ L−
+Iℓ×Iℓ, consider t × s as cluster in TIℓ+1×Iℓ+1 and
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+21
+(i) set
+˜gHL|t′×s′ = ˜gHL|t′×s′ + ΠHL|t′×s′gHℓ|t′×s′
+for all t′ × s′ ∈ children(t × s) with t′ × s′ ∈ L+
+Iℓ+1×Iℓ+1 by solving the local
+systems Equation (21),
+(ii) set
+˜gHℓ+1|t′×s′ = ˜gHℓ+1|t′×s′ + gHℓ|t′×s′
+for all t′×s′ ∈ children(t×s) with t′×s′ ∈ L−
+Iℓ+1×Iℓ+1 by dense matrix addition.
+Proof. We first list the computational cost of every step.
+Step 1: This step is without computational cost.
+Step 2: The computational cost for assembling {Qt}t∈TIL are bounded in Lemma 3.10, the
+ones for {Rt}t∈TIL and {Nt}t∈TIL in Lemma 5.1. The total cost of this step are thus
+6CH2C2
+minn2
+min(α + β)2δd|IL|.
+Step 3: We first list the computational cost for each substep for fixed ℓ.
+Step 3a: The computational cost are bounded in Lemma 5.2. Summing up the cost for this
+step yields
+4CH2C2
+minn2
+min
+(α(L − ℓ + 1) + β)3δd
+(α + β)δd
+|Iℓ|.
+Step 3b: The computational cost for solving Equation (20) are given by
+�
+r∈{s,t}
+�
+2
+�
+K(L)
+r
+�3
++
+�
+K(L)
+r
+��
+K(ℓ)
+r
+�2�
+≤ 3
+�
+r∈{s,t}
+�
+K(L)
+r
+�3
+and arise for all t × s ∈ L+
+Iℓ×Iℓ, while the efforts for Equation (21) are given by
+�
+r∈{s,t}
+�
+2
+�
+K(L)
+r
+�3
++
+�
+K(L)
+r
+�
+C2
+minn2
+min
+�
+≤ 3C2
+minn2
+min
+�
+r∈{s,t}
+�
+K(L)
+r
+�3
+and arise for all t × s ∈ L−
+Iℓ×Iℓ ∩ L+
+IL×IL.
+Step 3c: This substep is concerned with all t × s ∈ L−
+Iℓ×Iℓ \ L+
+IL×IL. Thus, a prolongation
+from Vhℓ|t ⊗ Vhℓ|s to Vhℓ+1|t ⊗ Vhℓ+1|s is required. This can be accomplished in at most
+2CuniC3
+minn3
+min operations.
+Step 3(c)i: For all t′ × s′ ∈ children(t × s) ∩ L+
+Iℓ+1×Iℓ+1 we need to solve Equation (21) on
+the level pair (L, ℓ + 1) instead of (L, ℓ), i.e.,
+Qs′u(L)
+s′×t′Q⊺
+t′ = N(L,ℓ+1)
+s′
+g(ℓ+1)
+s′×t′
+�
+N(L,ℓ+1)
+t′
+�⊺
+.
+The cost for a given t × s ∈ L−
+Iℓ×Iℓ \ L+
+IL×IL are thus bounded by
+�
+t′×s′∈children(t×s)
+�
+r∈{s′,t′}
+�
+2
+�
+K(L)
+r
+�3
++
+�
+K(L)
+r
+�
+C2
+minn2
+min
+�
+≤ 3C2
+minC2
+abn2
+min
+�
+r∈{s,t}
+�
+K(L)
+r
+�3
+.
+Step 3(c)ii:: The computational cost for this step are negligible.
+Steps 3b and 3c combined: Combining the preliminary considerations above and using
+Lemma A.7, the combined total computational cost for fixed ℓ for Step 3b and 3c are
+bounded by
+9CH2C2
+minC2
+abn2
+min
+(α(L − ℓ + 1) + β)3δd
+(α + β)δd
+|Iℓ| + 2CuniC3
+minn3
+min|Iℓ|.
+
+22
+J. D ¨OLZ
+Overall cost: Summing up the contributions of each step, yields that the overall cost of the
+algorithm are bounded by
+L−1
+�
+ℓ=0
+�
+19CH2C2
+minC2
+abn2
+min
+(α(L − ℓ + 1) + β)3δd
+(α + β)δd
++ 2CuniC3
+minn3
+min
+�
+|Iℓ|
+≤ |I0|
+L−1
+�
+ℓ=0
+�
+19CH2C2
+minC2
+abn2
+min
+(α(L − ℓ + 1) + β)3δd
+(α + β)δd
++ 2CuniC3
+minn3
+min
+�
+Cℓ
+uni
+≤ |I0|
+�
+19CH2C2
+minC2
+abn2
+min
+L−1
+�
+ℓ=0
+(α(L − ℓ + 1) + β)3δd
+(α + β)δd
+Cℓ
+uni + 2CuniC3
+minn3
+min
+CL
+uni − 1
+Cuni − 1
+�
+.
+We note that
+L−1
+�
+ℓ=0
+(α(L − ℓ + 1) + β)3δd
+(α + β)δd
+Cℓ
+uni ≤ CL
+uni
+L
+�
+ℓ=0
+((α + β) + αℓ)3δd
+(α + β)δd
+C−ℓ
+uni
+≤
+CL
+uni
+(α + β)δd
+∞
+�
+ℓ=0
+((β + α) + αℓ)⌈3δd⌉C−ℓ
+uni
+where
+∞
+�
+ℓ=0
+(β + αℓ)kqℓ ≤
+�
+1 +
+1
+1 − q
+�
+q
+1 − q + 1
+2
+�k
+k!
+�
+(α + β)k
+for all q ∈ [0, 1) and k ∈ N0 due to [4, Lemma 3.50 and 3.51]. The assertion follows with
+|I0|CL
+uni = |IL|.
+□
+Remark 5.4. The implementation effort for the H2-MLSCE estimator is comparatively low and
+along the lines of the usual H2-algorithms. In fact, given any H2-library, the H2-MLSCE estimator
+only requires the implementation of the three algorithms in Theorem 3.11, Definition 4.2, and
+Theorem 5.3. To that end, we remark that the initialization of {Qt}t∈TIL , {Rt}t∈TIL , {Nt}t∈TIL ,
+{R(L,ℓ)
+t
+}t∈TIℓ , and {N(L,ℓ)
+t
+}t∈TIℓ can algorithmically all be treated by the same subroutine.
+5.5. Computational work vs. accuracy. Combining Theorem 3.20 and Theorem 5.3 yields
+that the H2-MLSCE can be computed in O
+� �L
+ℓ=0 Mℓ|Iℓ|
+�
+operations, with δ entering only in the
+constant. Thus, it remains to choose the sample numbers such that accuracy of the finest level is
+achieved with minimal work. In complete analogy to various references, we mention [28, Appendix
+D] or [37] for example, we state the following theorem without proof.
+Theorem 5.5. Let the assumptions of Corollary 4.9 hold and choose ε > 0. The H2-MLSCE
+with
+L = d
+˜γ
+����
+log(ε−1)
+log(Cuni)
+����
+and sample numbers
+Mℓ = M0C−2ℓ(1+˜γ/d)/3
+uni
+,
+ℓ = 0, . . . , L,
+with
+M0 =
+�
+�
+�
+�
+�
+C2˜γL/d
+uni
+for 2˜γ > d,
+C2˜γL/d
+uni
+L2
+for 2˜γ = d,
+C2(1+˜γ/d)L/3
+uni
+for 2˜γ < d,
+achieves error estimates
+��g − EML
+L
+[ΠH
+h,Lg]
+��
+L2
+P(Ω;L2(D×D)) = O(ε)
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+23
+Figure 5. Sample realizations of the centered Gaussian process with G3/2-
+asymptotically smooth covariance function taken for the numerical experiments.
+and
+sup
+uh,vh∈VhL
+���
+�
+D
+�
+D
+�
+g(x, y) − ΠHLEML
+L
+[Πmix
+hL g(x, y)]
+�
+uh(x)vh(y) dµ(x) dµ(y)
+���
+L2
+P(Ω)
+∥uh∥L2(D)∥vh∥L2(D)
+= O(ε)
+in a computational complexity of
+�
+�
+�
+�
+�
+O(ε−2)
+for 2˜γ > d,
+O
+�
+ε−2| log(ε−1)|3�
+for 2˜γ = d,
+O(ε−d/˜γ)
+for 2˜γ < d.
+Thus, for 2˜γ > d, the overall error is dominated by the Monte Carlo error, whereas for 2˜γ < d
+the overall error is dominated by the error of the approximation spaces Vhl.
+We note that these computational complexities are in line with the wavelet-based approach
+from [28], but the H2-approach does not require a hierarchical basis. In contrast, wavelet-based
+approaches are theoretically also applicable if the smoothness of the kernel function is finite, which
+is, see also Remark 2.10, asymptotically not the case for the H2-approach due to the increasingly
+higher polynomial degrees required for interpolation.
+6. Numerical experiments
+For our numerical experiments we aim at estimating the covariance of a Gaussian random
+field at the surface ∂D of a turbine geometry, see Figure 5, i.e., on a two-dimensional manifold
+embedded into R3. The radius of the turbine to the end of the blades is 1.5. To that end, we
+prescribe a reference Gaussian random field in terms of a Karhunen-Lo´eve expansion, i.e.,
+Z(ω, x) =
+∞
+�
+k=0
+�
+λkϕ(x)Yk(ω),
+
+2.5e+00
+.5
+0.5
+-0.5
+1.5
+-2.5e+002.5e+00
+.5
+0.5
+-0.5
+1.5
+-2.5e+002.5e+00
+.5
+Q.5
+-0.5
+-1.5
+-2.5e+002.5e+00
+.5
+0.5
+-0.5
+-1.5
+-2.5e+0024
+J. D ¨OLZ
+L
+0
+1
+2
+3
+4
+5
+6
+dim Vh = dim Wh
+60
+240
+960
+3 840
+15 360
+61 440
+245 760
+dim(Wh ⊗ Wh)
+3 600
+57 600
+921 600
+≈ 14.7 · 106
+≈ 236 · 106
+≈ 3.77 · 109
+≈ 60.4 · 109
+Table 1.
+Dimensions of the used finite element spaces. The estimated covari-
+ance matrices are matrices in Rdim Wh×dim Wh, i.e., have dim(Wh ⊗ Wh) degrees
+of freedom.
+L
+0
+1
+2
+3
+4
+5
+6
+M0
+1
+4
+64
+576
+4 096
+25 600
+147 456
+M1
+1
+16
+144
+1 024
+6 400
+36 864
+M2
+4
+36
+256
+1 600
+9 216
+M3
+9
+64
+400
+2 304
+M4
+16
+100
+576
+M5
+25
+144
+M6
+36
+Table 2. Sample numbers chosen according to the case 2˜γ = d in Theorem 5.5
+for the numerical example.
+with Yk ∼ U([−1, 1]) and {(λk, ϕk)}∞
+k=0 the eigenpairs of the integral operator
+C : L2(∂D) → L2(∂D),
+(Cϕ)(x) =
+�
+∂D
+gδ(x, y)ϕ(y) dσ(y).
+The covariance function gδ is chosen as a modified Mat´ern-9/2 kernel
+gδ(x, y) = ˜g(∥γδ(x) − γδ(y)∥),
+˜g(r) =
+�
+1 + 3r + 27r2
+7
++ 18r3
+7
++ 27r4
+35
+�
+e−3r,
+where
+γδ : ∂D → R3,
+γδ(x1, x2, x3), =
+�
+�
+0.1 + Υδ(2 ∗ x1 − 1)x1
+x2
+x3
+�
+�
+and
+Υδ(t) =
+υδ(1 − t)
+υδ(1 − t) + υδ(t),
+υδ(t) =
+�
+0,
+t ≤ 0,
+e−t
+1
+1−δ ,
+t > 0,
+is a partition of Gevrey class δ ≥ 1 with Υ(t) = 1 for t < 0 and Υ(t) = 0 for t > 0, see, e.g., [13].
+For our numerical experiments we choose δ = 3/2, for which samples are illustrated in Figure 5.
+This makes the covariance function gδ a G3/2-asymptotically smooth kernel function.
+The H2-implementation of the numerical experiments is based on the C++-Library Bembel [17],
+with compression parameters α = 1, β = 2, η = 0.8, and nmin = 4. We choose piecewise constant
+finite element spaces Vhℓ = Whℓ, ℓ = 0, 1, 2, . . ., on uniformly refined quadrilateral meshes with
+Cuni = 4 and hℓ ∼ 2−ℓ, leading to dimensions of the finite element spaces and covariance matrices
+as in Table 1. The Gaussian random field samples ΠhℓZ are generated from a Karhunen Lo´eve
+expansion which is truncated at 10−3hℓ and computed from a pivoted Cholesky decomposition [30].
+According to Corollary 4.10 and Theorem 5.5 it holds ˜γ = 1 and we can expect a linear convergence
+rate for our H2-MLSCE, if the sample numbers are chosen proportional to Theorem 5.5. For our
+particular example we choose the sample numbers listed in Table 2. Figure 6 shows that we reach
+indeed the predicted rate convergence rate of Theorem 4.8 and a computational work vs. accuracy
+as in Theorem 5.5. The spectral error was computed with a power iteration up to an absolute
+accuracy of 10−4. The computation times are measured in wall clock time and have been carried
+out in parallel with 48 threads on a compute server with 1.3TB RAM and two Intel(R) Xeon(R)
+CPU E7-4850 v2 CPUs with Hyper-Threading enabled.
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+25
+0
+2
+4
+6
+100
+101
+102
+L
+absolute spectral error
+Convergence
+˜γ = 1
+100
+101
+102
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+103
+104
+105
+106
+ε
+wall clock time (sec)
+Computational work vs. accuracy
+ε−2| log(ε)|3
+Figure 6. Convergence plot of a realization of the H2-MLSCE and corresponding
+computational work vs. accuracy with the sample numbers as in Table 2, cf. also
+Corollary 4.10 and Theorem 5.5.
+7. Conclusion
+In this article, we considered the multilevel estimation of covariance functions which are Gδ-
+asymptotically smooth, δ ≥ 1.
+This choice is motivated by the stochastic partial differential
+equation approach to Gaussian random fields and pseudodifferential operator theory. The naive
+approach to estimate the covariance function from discretized samples using the single level covari-
+ance estimator is computationally prohibitive due to the density of the arising covariance matrices
+and the slow convergence of the sample covariance estimator. To overcome these issues, we first
+generalized the classical H2-approximation theory for asymptotically smooth kernels to Gevrey
+kernels. This allows to compress the arising covariance matrices by H2-matrices in linear com-
+plexity with respect to the underlying approximation space. Secondly, we proposed and analyzed
+an H2-formatted multilevel covariance sample estimator (H2-MLCSE). This estimator exploits an
+approximate multilevel hierarchy in the H2-approximation spaces to estimate the covariance in
+the same complexity as the mean. The provided approximation theory is applicable to a rather
+general setting, covering for example domains, manifolds, graphs, and multi-screens as well as
+various approximation spaces such as finite element spaces and Nystr¨om discretizations.
+Alternatively to the approach proposed in this paper, a wavelet based method for estimating
+covariance functions was proposed in [28]. The advantage of such a wavelet method is that the
+wavelet-based approximation results also hold for finite smoothness of the covariance function,
+whereas the here presented H2-approach requires asymptotically in��nite smoothness. In contrast,
+the advantage of the H2-approach in this paper is that no wavelet basis is required and that the
+presented algorithms can be integrated into the many readily available H2-matrix codes.
+Acknowledgement
+The author would like to express his sincere gratitude to Christoph Schwab for the initial
+discussions on generalizing the H2-matrix approximation theory to Gevrey kernels and for critical
+and helpful comments during the writing of the manuscript.
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+Appendix A. Computation of H2-related constants
+Definition A.1 ([4, Definition 3.44]). Let TI be a cluster tree and denote the number of interpo-
+lation points chosen in each cluster t ∈ TI by Kt. We say that {Kt}t∈TI is a rank distribution.
+We say that {Kt}t∈TI is a (Cbn, α, β, r, ξ)-bounded rank distribution, Cbn ≥ 1, α > 0, β ≥ 0,
+r ≥ 1, ξ ≥ 1, if
+���
+t ∈ TI : Kt > (α + β(ℓ − 1))r��� ≤ Cbnξ−ℓ|TI|,
+for all ℓ ∈ N.
+Lemma A.2. Let TI be a cluster tree on the index set I satisfying Assumption 2.12.
+Then
+{Kt}t∈TI is a (1, α, β, δd, Cab)-bounded rank distribution if the number of interpolation points in
+(Kt)t∈TI are chosen according to Equation (8), i.e.,
+Kt =
+�
+(β + α(p − level(t)))δ�d
+Proof. The proof is analogy to the example in [4, p. 64]. Let p denote the depth of TI. We need
+to bound the number of clusters with
+Kt =
+�
+(β + α(p − level(t)))δ�d ≥ (β + α(p − level(t)))δd > (α + β(ℓ − 1))δd.
+From this inequality we deduce that the clusters satisfying this constraint also satisfy level(t) <
+p + 1 − ℓ. Due to Assumption 2.12 the number of such clusters is bounded by from above by
+(Cp−ℓ+2
+ab
+− 1)/(Cab − 1) and we obtain the assertion due to
+|TI| ≥ Cp+2
+ab
+− 1
+Cab − 1 = Cℓ
+ab
+Cp−ℓ+2
+ab
+− C−ℓ
+ab
+Cab − 1
+≥ Cℓ
+ab
+Cp−ℓ+2
+ab
+− 1
+Cab − 1
+.
+□
+
+28
+J. D ¨OLZ
+Definition A.3 ([4, Definitions 3.43 and 3.47]). Let TI be a cluster tree.
+We say that it is
+(Crc, α, β, r, ξ)-bounded with Crc ≥ 1, α > 0, β ≥ 0, r ≥ 1, ξ > 1, if
+(22)
+���
+t ∈ LI : |t| > (β + α(ℓ − 1))r��� ≤ Crcξ−ℓ|TI|,
+for all ℓ ∈ N,
+and
+| children(t)| ≤ Crc,
+for all t ∈ TI.
+(23)
+We say that TI is (Crc, α, β, r, ξ)-regular, if it is (Crc, α, β, r, ξ)-bounded and additionally satisfies
+| children(t)| ≥ 2,
+for all t ∈ TI \ LI,
+(24)
+(α + β)r ≤ Crc|t|,
+for all t ∈ LI.
+(25)
+Lemma A.4. Let TI be a cluster tree with depth p on the index set I satisfying Assumption 2.12.
+Then TI is (Crc, α, β, δd, Cab)-regular with
+(26)
+Crc = max
+�
+Cab, (α + β)δd
+nmin
+, C
+n1/(δd)
+min
+−β+α
+α
++1
+ab
+�
+.
+Proof. Equation (6) implies 2 ≤ | children(t)| ≤ Cab, t ∈ TI \ LI, which yields (24) and Equa-
+tion (23) holds with Crc ≥ Cab. Inserting the upper bound from Equation (7) into Equation (25)
+yields
+(α + β)δd
+nmin
+≤ Crc.
+Finally, the lower bound from Equation (7) implies that there are at most Cp+1
+ab
+leafs. The
+upper bound from Equation (7) and Equation (22) with ξ = Cab then imply that Crc must satisfy
+Crc ≥
+� Cp+ℓ+1
+ab
+|TI|
+for all ℓ with (β + α(ℓ − 1))δd < nmin
+0
+else
+Solving (β + α(ℓ − 1))δd < nmin for ℓ implies ℓ < (n1/(δd)
+min
+− β + α)/α which yields
+C
+n1/(δd)
+min
+−β+α
+α
++1
+ab
+≤ Crc
+due to |TI| ≥ (Cp+2
+ab
+− 1)/(Cab − 1). Combining all conditions on Crc yields the assertion.
+□
+Lemma A.5 ([4, Lemma 3.45]). Let TI be a (Crc, α, β, r, ξ)-bounded cluster tree and let {Kt}t∈TI
+be a (Cbn, α, β, r, ξ)-bounded rank distribution. Define
+kt =
+�
+max{Kt, |t|},
+t ∈ LI,
+max{Kt, �
+t′∈children(t) Kt′},
+t ∈ TI \ LI.
+(27)
+and m ∈ N. Then there is a constant Ccb = Ccb(Crc, Cbn, r, ξ) ≥ 1 such that
+�
+t∈TI
+km
+t ≤ Ccb(α + β)rm|TI|.
+Lemma A.6 ([4, Lemma 3.48]). Let TI be a (Crc, α, β, r, ξ)-regular cluster tree. Then it holds
+|TI| ≤ 2Crc|I|
+(α + β)r
+Lemma A.7 (Modification of [4, Corollary 3.49]). Let TI be (Crc, α, β, r, ξ)-bounded and (Kt)t∈TI
+be a (Cbn, α, β, ξ)-bounded rank distribution.
+Let TI be (Crc, α′, β′, r, ξ)-regular and TI×I be a
+block-cluster tree with sparsity constant Csp. For m ∈ N and {kt}t∈TI defined as in Equation (27)
+it holds
+�
+t∈TI
+km
+t ≤ CH2 (α + β)rm
+(α′ + β′)r |I|
+with CH2 = 2CrcCcb.
+Proof. Combine Lemma A.5 and Lemma A.6.
+□
+
+DATA SPARSE MULTILEVEL COVARIANCE ESTIMATION IN OPTIMAL COMPLEXITY
+29
+Institute for Numerical Simulation, University of Bonn, Friedrich-Hirzebruch-Allee 7, 53115 Bonn,
+Germany
+Email address: [email protected]
+