Daily Papers

We propose to study and promote the robustness of a model as per its performance through the interpolation of training data distributions. Specifically, (1) we augment the data by finding the worst-case Wasserstein barycenter on the geodesic connecting subpopulation distributions of different categories. (2) We regularize the model for smoother performance on the continuous geodesic path connecting subpopulation distributions. (3) Additionally, we provide a theoretical guarantee of robustness improvement and investigate how the geodesic location and the sample size contribute, respectively. Experimental validations of the proposed strategy on four datasets, including CIFAR-100 and ImageNet, establish the efficacy of our method, e.g., our method improves the baselines' certifiable robustness on CIFAR10 up to 7.7%, with 16.8% on empirical robustness on CIFAR-100. Our work provides a new perspective of model robustness through the lens of Wasserstein geodesic-based interpolation with a practical off-the-shelf strategy that can be combined with existing robust training methods.

In this paper we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud to obtain lower dimensional bundles. Local approximations obtained by local PCAs are collected into a rank k tangent subbundle on R^d, k<d, which we call a principal subbundle. This determines a sub-Riemannian metric on R^d. We show that sub-Riemannian geodesics with respect to this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold M, construction of a representation of the point-cloud in R^k, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, the framework generalizes to observations on an a priori known Riemannian manifold.

By interpreting the product of the Principal Component Analysis, that is the covariance matrix, as a sequence of nested subspaces naturally coming with weights according to the level of approximation they provide, we are able to embed all d--dimensional Grassmannians into a stratified space of covariance matrices. We observe that Grassmannians constitute the lowest dimensional skeleton of the stratification while it is possible to define a Riemaniann metric on the highest dimensional and dense stratum, such a metric being compatible with the global stratification. With such a Riemaniann metric at hand, it is possible to look for geodesics between two linear subspaces of different dimensions that do not go through higher dimensional linear subspaces as would euclidean geodesics. Building upon the proposed embedding of Grassmannians into the stratified space of covariance matrices, we generalize the concept of varifolds to what we call flagfolds in order to model multi-dimensional shapes.

Symmetric Positive Definite (SPD) matrices are ubiquitous in data analysis under the form of covariance matrices or correlation matrices. Several O(n)-invariant Riemannian metrics were defined on the SPD cone, in particular the kernel metrics introduced by Hiai and Petz. The class of kernel metrics interpolates between many classical O(n)-invariant metrics and it satisfies key results of stability and completeness. However, it does not contain all the classical O(n)-invariant metrics. Therefore in this work, we investigate super-classes of kernel metrics and we study which key results remain true. We also introduce an additional key result called cometric-stability, a crucial property to implement geodesics with a Hamiltonian formulation. Our method to build intermediate embedded classes between O(n)-invariant metrics and kernel metrics is to give a characterization of the whole class of O(n)-invariant metrics on SPD matrices and to specify requirements on metrics one by one until we reach kernel metrics. As a secondary contribution, we synthesize the literature on the main O(n)-invariant metrics, we provide the complete formula of the sectional curvature of the affine-invariant metric and the formula of the geodesic parallel transport between commuting matrices for the Bures-Wasserstein metric.

A point of a metric space is called a k-hub if it is the endpoint of exactly k disjoint geodesics, and that the concatenation of any two of these paths is still a geodesic. We prove that in the Brownian sphere, there is no k-hub for kgeq 3.

We introduce a principled way of computing the Wasserstein distance between two distributions in a federated manner. Namely, we show how to estimate the Wasserstein distance between two samples stored and kept on different devices/clients whilst a central entity/server orchestrates the computations (again, without having access to the samples). To achieve this feat, we take advantage of the geometric properties of the Wasserstein distance -- in particular, the triangle inequality -- and that of the associated {\em geodesics}: our algorithm, FedWad (for Federated Wasserstein Distance), iteratively approximates the Wasserstein distance by manipulating and exchanging distributions from the space of geodesics in lieu of the input samples. In addition to establishing the convergence properties of FedWad, we provide empirical results on federated coresets and federate optimal transport dataset distance, that we respectively exploit for building a novel federated model and for boosting performance of popular federated learning algorithms.

Although data diffusion embeddings are ubiquitous in unsupervised learning and have proven to be a viable technique for uncovering the underlying intrinsic geometry of data, diffusion embeddings are inherently limited due to their discrete nature. To this end, we propose neural FIM, a method for computing the Fisher information metric (FIM) from point cloud data - allowing for a continuous manifold model for the data. Neural FIM creates an extensible metric space from discrete point cloud data such that information from the metric can inform us of manifold characteristics such as volume and geodesics. We demonstrate Neural FIM's utility in selecting parameters for the PHATE visualization method as well as its ability to obtain information pertaining to local volume illuminating branching points and cluster centers embeddings of a toy dataset and two single-cell datasets of IPSC reprogramming and PBMCs (immune cells).

Pre-trained multi-modal models, such as CLIP, provide transferable embeddings and show promising results in diverse applications. However, the analysis of learned multi-modal embeddings is relatively unexplored, and the embedding transferability can be improved. In this work, we observe that CLIP holds separated embedding subspaces for two different modalities, and then we investigate it through the lens of uniformity-alignment to measure the quality of learned representation. Both theoretically and empirically, we show that CLIP retains poor uniformity and alignment even after fine-tuning. Such a lack of alignment and uniformity might restrict the transferability and robustness of embeddings. To this end, we devise a new fine-tuning method for robust representation equipping better alignment and uniformity. First, we propose a Geodesic Multi-Modal Mixup that mixes the embeddings of image and text to generate hard negative samples on the hypersphere. Then, we fine-tune the model on hard negatives as well as original negatives and positives with contrastive loss. Based on the theoretical analysis about hardness guarantee and limiting behavior, we justify the use of our method. Extensive experiments on retrieval, calibration, few- or zero-shot classification (under distribution shift), embedding arithmetic, and image captioning further show that our method provides transferable representations, enabling robust model adaptation on diverse tasks. Code: https://github.com/changdaeoh/multimodal-mixup

Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology and physics. While it is thought that these methods preserve underlying manifold structure of data by learning a proxy for geodesic distances, no specific theoretical links have been established. Here, we establish such a link via results in Riemannian geometry explicitly connecting heat diffusion to manifold distances. In this process, we also formulate a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. This novel perspective makes clearer the choices available in manifold learning and denoising. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances, and preserving cluster structure in toy datasets. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure, where our method enables interpolation of withheld timepoints of data. Finally, we show that parameters of our more general method can be configured to give results similar to PHATE (a state-of-the-art diffusion based manifold learning method) as well as SNE (an attraction/repulsion neighborhood based method that forms the basis of t-SNE).

One of the primary challenges in brain tumor segmentation arises from the uncertainty of voxels close to tumor boundaries. However, the conventional process of generating ground truth segmentation masks fails to treat such uncertainties properly. Those "hard labels" with 0s and 1s conceptually influenced the majority of prior studies on brain image segmentation. As a result, tumor segmentation is often solved through voxel classification. In this work, we instead view this problem as a voxel-level regression, where the ground truth represents a certainty mapping from any pixel to the border of the tumor. We propose a novel ground truth label transformation, which is based on a signed geodesic transform, to capture the uncertainty in brain tumors' vicinity. We combine this idea with a Focal-like regression L1-loss that enables effective regression learning in high-dimensional output space by appropriately weighting voxels according to their difficulty. We thoroughly conduct an experimental evaluation to validate the components of our proposed method, compare it to a diverse array of state-of-the-art segmentation models, and show that it is architecture-agnostic. The code of our method is made publicly available (https://github.com/Oulu-IMEDS/SiNGR/).

In molecular dynamics (MD) simulations, rare events, such as protein folding, are typically studied by means of enhanced sampling techniques, most of which rely on the definition of a collective variable (CV) along which the acceleration occurs. Obtaining an expressive CV is crucial, but often hindered by the lack of information about the particular event, e.g., the transition from unfolded to folded conformation. We propose a simulation-free data augmentation strategy using physics-inspired metrics to generate geodesic interpolations resembling protein folding transitions, thereby improving sampling efficiency without true transition state samples. Leveraging interpolation progress parameters, we introduce a regression-based learning scheme for CV models, which outperforms classifier-based methods when transition state data is limited and noisy

Large-scale language-vision pre-training models, such as CLIP, have achieved remarkable text-guided image morphing results by leveraging several unconditional generative models. However, existing CLIP-guided image morphing methods encounter difficulties when morphing photorealistic images. Specifically, existing guidance fails to provide detailed explanations of the morphing regions within the image, leading to misguidance. In this paper, we observed that such misguidance could be effectively mitigated by simply using a proper regularization loss. Our approach comprises two key components: 1) a geodesic cosine similarity loss that minimizes inter-modality features (i.e., image and text) on a projected subspace of CLIP space, and 2) a latent regularization loss that minimizes intra-modality features (i.e., image and image) on the image manifold. By replacing the na\"ive directional CLIP loss in a drop-in replacement manner, our method achieves superior morphing results on both images and videos for various benchmarks, including CLIP-inversion.

The recent video grounding works attempt to introduce vanilla contrastive learning into video grounding. However, we claim that this naive solution is suboptimal. Contrastive learning requires two key properties: (1) alignment of features of similar samples, and (2) uniformity of the induced distribution of the normalized features on the hypersphere. Due to two annoying issues in video grounding: (1) the co-existence of some visual entities in both ground truth and other moments, \ie semantic overlapping; (2) only a few moments in the video are annotated, \ie sparse annotation dilemma, vanilla contrastive learning is unable to model the correlations between temporally distant moments and learned inconsistent video representations. Both characteristics lead to vanilla contrastive learning being unsuitable for video grounding. In this paper, we introduce Geodesic and Game Localization (G2L), a semantically aligned and uniform video grounding framework via geodesic and game theory. We quantify the correlations among moments leveraging the geodesic distance that guides the model to learn the correct cross-modal representations. Furthermore, from the novel perspective of game theory, we propose semantic Shapley interaction based on geodesic distance sampling to learn fine-grained semantic alignment in similar moments. Experiments on three benchmarks demonstrate the effectiveness of our method.