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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf,len=694
+page_content='Deep Axial Hypercomplex Networks  Nazmul Shahadat, Anthony S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Maida  University of Louisiana at Lafayette  Lafayette LA 70504, USA  nazmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='ruet@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='com, maida@louisiana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='edu  Abstract  Over the past decade, deep hypercomplex-inspired  networks have enhanced feature extraction for image  classification by enabling weight sharing across input  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Recent works make it possible to improve  representational capabilities by using hypercomplex-  inspired networks which consume high computational  costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  This paper reduces this cost by factorizing  a quaternion 2D convolutional module into two con-  secutive vectormap 1D convolutional modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Also,  we use 5D parameterized hypercomplex multiplication  based fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Incorporating both yields  our proposed hypercomplex network, a novel architec-  ture that can be assembled to construct deep axial-  hypercomplex networks (DANs) for image classifica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' We conduct experiments on CIFAR benchmarks,  SVHN, and Tiny ImageNet datasets and achieve bet-  ter performance with fewer trainable parameters and  FLOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Our proposed model achieves almost 2% higher  performance for CIFAR and SVHN datasets, and more  than 3% for the ImageNet-Tiny dataset and takes six  times fewer parameters than the real-valued ResNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Also, it shows state-of-the-art performance on CIFAR  benchmarks in hypercomplex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Introduction  Convolutional neural networks (CNNs) and hyper-  complex CNNs (HCNNs) for image classification form  a hierarchical design where different layers extract dif-  ferent levels of feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  CNNs have  shown significant success in recent decades [2, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' In  vision tasks, these CNN-based feature extraction de-  signs can be improved in regard to working with multi-  dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' To enhance the CNNs ability, HC-  NNs have been used which treat the multi-dimensional  data as a cohesive entity by applying cross-channel  weight sharing to discover cross-channel relationships  [4, 5, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Also, implementations in hypercomplex  space provide more advantages [1, 3, 7, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' It has also  been shown that the HCNNs could create better output  representations [13,16,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Recently, HCNNs with various dimensions like 2D  HCNNs [21], 4D HCNNs [4, 14, 15], 8D HCNNs [20],  or generalized HCNNs [5], have been studied and have  hypercomplex properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The reason behind the suc-  cess of HCNNs is that they capture the cross-channel  relationships [4,5,14,15,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Among them, quaternion  networks have a set of algebra operations and they have  outperformed than the other HCNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Stacking quater-  nion convolutional coherent layers have achieved better  representational feature maps and have shown promis-  ing results in vision tasks [4,15,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These networks are  cost-effective compared to real-valued CNNs and fully  connected networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' But still, they are very expensive  for large inputs like vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  This work uses an axial hypercomplex network that:  1) handles multidimensional inputs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 2) applies weight  sharing across input channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 3) captures cross-channel  correlations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 4) reduces computational costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' and 5) in-  creases validation accuracy performance for image clas-  sification datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  The main idea of this work is to  decompose hypercomplex 2D convolutional operation  into two consecutive vectormap 1D convolutional oper-  ations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' By splitting 2D spatial convolution operation  into height-axis and width-axis spatial convolution, it  enables the model to reduce cost once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Addition-  ally, we apply a quaternion-based stem layer, and pa-  rameterized hypercomplex multiplication (PHM) based  fully connected layer to get better representation and  better generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  This paper conducts extensive experiments that show  the effectiveness of our novel axial hypercomplex net-  works on four image classification datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Our novel  contribution is a new model that factorizes the two-  dimensional spatial hypercomplex convolutional oper-  ation into two one-dimensional operations along the  height-axis and width-axis sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Our contribu-  tions are:  Replacing the spatial 3 × 3 QCNN in the bottle-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='04626v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='CV]  11 Jan 2023  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Proposed axial-hypercomplex network with PHM-based fully-connected layer in backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' “AHNN” stands for axial-  hypercomplex neural network bottleneck block which is described in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Here, Qin = Qr + Qw + Qx + Qy + Qz,  H = Hr + Hw + Hx + Hy + Hz, and Qout = Qro + Qwo + Qxo + Qyo + Qzo are the input, hypercomplex parameterized  weight, and output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' For the calculation of H see the “PHM Layer” section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  neck block of quaternion ResNets using two VC-  NNs and showing the effectiveness of the proposed  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Applying QCNN in the stem layer (the first layer of  the network), resulting in a quaternion-stem model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Like QPHM [16], applying PHM-based dense layer  in the backend of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  This proposed axial hypercomplex ResNets outper-  formed the baseline networks for classification datasets  which is shown in Tables 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Our experiments  show that the proposed model achieves state-of-the-art  results with far fewer trainable parameters, and FLOPS  for CIFAR benchmarks in hypercomplex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Background and Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Quaternion Convolution  The deep quaternion CNN extends of complex CNNs  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' This section explains cross channel weight sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  [4] and [15] extended the principles of quaternion con-  volution operations, and weight initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Quater-  nion number system is formed as, Q = r + ix + jy +  kz ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' r, x, y, z ∈ R where, r, x, y, and z are real values  and i, j, and k are imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Quaternion convolution  between quaternion filter matrix F and quaternion input  vector M, is defined as [4]:  M ⊛ F = (Or, Oi, Oj, Ok)  = (Mr ∗ Fr − Mi ∗ Fi − Mj ∗ Fj − Mk ∗ Fk,  Mi ∗ Fr + Mr ∗ Fi + Mj ∗ Fk − Mk ∗ Fj,  Mj ∗ Fr + Mr ∗ Fj + Mk ∗ Fi − Mi ∗ Fk,  Mk ∗ Fr + Mr ∗ Fk + Mi ∗ Fj − Mj ∗ Fi)  (1)  where, M ⊛ F, and all others are quaternion numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Or is the real part, and Oi, Oj, and Ok are the imag-  inary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Although there are 16 real-valued convo-  lutions in Equation 1, there are only four kernels that  are reused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The weight sharing happens this way [14]  which forces the model to learn cross-channel interre-  lationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' According to the quaternion definition, a  quaternion layer can accept four or m numbers of input  channels, where m is divisible by four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' To process m  input channels (m ≥ 4), m/4 number of independent  quaternion convolution modules is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Also, there  are m/4 weight sets where each module has its own  weight sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Cross-channel weight sharing allows dis-  covering of cross-channel input correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Our weight  initialization was the same as [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Vectormap Convolution  We explain 3D generalized hypercomplex networks  or VCNNs as VCNNs are used in our proposed mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The VCNN is more flexible as it doesn’t require 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Parameterized  STAGE 1  STAGE 2  STAGE 3  STAGE 4  Weight  Input  Horse  Cat  224x224  Classification  PHM  Stem Layer  based FC Layer  QCNN  64 Filters  1st Layer  2nd Layer  3rd Layer  4rth Layer  AHNN  AHNN  AHNN  Flatten  Filter size 7  AHNN  128 Filters  256 Filters  512 Filters  with stride 2  64 Filters  Layer  max-pooling  Stride 2  Stride 2  Stride 2  Stride 1  5-dimensional PHM laye  VPHM  TimerFigure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' AHNN bottleneck block used in our proposed axial-hypercomplex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' “bn”, “quat”, and “VCNN” stand for batch  normalization, quaternion CNN, and vectormap CNN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  However, still using cross channel weight sharing this is  seen in 3 × 3 matrix used in Equation 2, only three fil-  ters A, B, and C are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The Vectormap convolution  operation is defined as:  �  �  R(M ∗ F)  I(M ∗ F)  J (M ∗ F)  �  � = L ⊙  �  �  A  B  C  C  A  B  B  C  A  �  � ∗  �  �  x  y  z  �  �  (2)  where, A, B, and C are real-valued kernels, x, y, and  z being real-valued vectors, and L is a learnable matrix,  L ∈ RD3×D3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' where D3 stands for 3-dimensional input  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The initial value of this matrix L is defined as:  L =  �  �  1  1  1  1  1  1  1  1  1  �  �  (3)  Our weight initialization follows [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' PHM Layer  Parameterized hypercomplex multiplication is an-  other form of generalized hypercomplex network, ex-  plained in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  As we use this PHM layer only in  the fully connected (FC) layer, our explanation is re-  stricted to this PHM-based dense layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' It is defined as,  y = Hx + b, where H ∈ Rk×d represents the PHM  layer and it is calculated as, H = �n  i=1 Ii ⊗ Ai, where  Ii ∈ Rn×n and Ai ∈ Rk/n×d/n are learnable parameter  matrices and i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' n (n = 4 or 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These matrices  can be reused which leads to parameter reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Also,  the ⊗ represents the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The flattened  layer which is the output of the CNN network is used as  an input to the PHM FC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These inputs are split as,  Qin = Qr + Qw + Qx + Qy + Qz and the outputs are  merged into Qout as, Qout = Qro+Qwo+Qxo+Qyo+  Qzo for 5D hypercomplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The 4D hypercomplex pa-  rameter matrix is discussed in [22] which expresses the  Hamiltonian product, and the 5D hypercomplex param-  eter matrix of PHM operation is explained in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' This  5D parameter matrix is used to construct a 5D PHM FC  layer which preserves all properties of the PHM layer  and hypercomplex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' This work uses 5D PHM  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Proposed Axial Hypercomplex Networks  Complex convolutional neural networks (CCNNs),  QCNNs, Octonions convolutional neural networks (OC-  NNs), VCNNs, and PHM are the versions of HCNNs  that provide all advantages of HCNNs like weight shar-  ing across input channels, and the ability to discover  cross channel correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These HCNNs perform bet-  ter with fewer trainable parameters for vision applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' But, they are still computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' For  vision tasks, these HCNNs take O(N 2) resources for  an image of length N where N is the flattened pixel  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' For a 2D image of height h and width w, where  N  = hw, and h = w, the computational cost is  O((hw)2) = O(h2w2) = O(h4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  This  section  describes  our  proposed  axial-  hypercomplex model in Figures 1 and 2 to reduce  the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Axial networks were first used  in [8, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  To implement our proposed model, we  followed the assumption that images are approximately  square where the pixel count of h and w are the same,  and both are much less than the pixel count of hw [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  To translate a quaternion convolutional bottleneck block  to an axial-hypercomplex bottleneck block, we replace  the 3 × 3 spatial quaternion convolutional operation  by two axial vectormap convolutional neural network  (VCNN) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These layers are applied to the height  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  b  rel  bn  3x1 VCNN  Width-Axis  enb  nb  relu  1  X  X  X  S  H  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  S  channel  s  sLayer  Output  size  Deep Quaternion  ResNet  Vectormap  ResNet  QPHM  Axial  Hypercomplex  Stem  32x32  3x3Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' std=1  3x3V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' std=1  3x3Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' std=1  3x3Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' std=1  Bottleneck  group 1  32x32  �  �  1x1Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' 480  �  �×3  �  �  1x1V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' 480  �  �×3  �  ���  1x1Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 120  3x1AV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 120  1x3AV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' 480  �  ���×3  Bottleneck  group 2  16x16  �  �  1x1Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' 960  �  �×4  �  �  1x1V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' 960  �  �×4  �  �  1x1QP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' 1920  �  �×6  �  �  1x1V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' 1920  �  �×6  �  ���  1x1Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' 960  3x3V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 960  1x1V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 3840  �  �×3  �  �  1x1QP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 960  3x3QP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 960  1x1QP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 3840  �  �×3  �  ���  1x1Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 960  3x1AV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 960  1x3AV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 960  1x1Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 3840  �  ���×3  Pooling layer  1x1x100  global average-pool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' 100 outputs  Output  1x1x100  fully connected layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' softmax  5D PHM layer  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The 50-layer architectures tested on CIFAR-100: quaternion ResNet [4, 5], vectormap ResNet [5], QPHM [16], and our  proposed axial-hypercomplex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Input is a 32x32x3 color image for CIFAR benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The number of stacked bottleneck  modules is specified by multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' “Q”, “V”, “QP”, “AV” and “std” denote quaternion convolution, 3D vectormap convolution,  QPHM (quaternion networks with 4D PHM layer), axial vectormap convolution, and stride correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Integers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=', 120,  240) denote the number of output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' PHM layer stands for parameterized hypercomplex multiplication layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' This work  uses 5D PHM based FC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  axis (3 channels 3x1 VCNN layer) and width axis (3  channels 1x3 VCNN layer) sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The two 1 × 1  quaternion convolutional layers remain unchanged  like the original QCNNs [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The 1 × 1 QCNNs are  responsible to reduce and then increase the number of  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' This forms our proposed axial-hypercomplex  bottleneck block seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  This block is  stacked multiple times to construct axial-hypercomplex  ResNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Axial-hypercomplex models only work on one di-  mension at a time but the input images are 2-  dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' For two-dimensional vision tasks, a square  2D input where h = w, so w2 = N, where N is the se-  quence length of the flattened pixel set, is split into two  1D vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The 3-channel VCNN operation is first ap-  plied along the 1D input image region of length h and  then applied along the 1D input image region of length  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These two 1D operations finally merged together re-  duces cost to O(h · h2) = O(h3) from the HCNNs cost  of O(h4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Each quaternion convolution accepts four channels  of input and produces four channels of output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Hence,  the required number of 1 × 1 quaternion conv2d mod-  ules equals the number of input channels divided by  four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The set of output channels of down-sampled 1 × 1  quaternion is merged into input to the axial VCNN mod-  ules, and the output channels of axial VCNN modules  are split into groups of four again for 1 × 1 up-sampled  quaternion conv2d layer [4,17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' One quaternion 2D con-  volution is applied to each group of four channels and  one vectormap 2D convolution is applied to each group  of three channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Like vectormap, each axial vectormap  module takes three input channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Thus, the weight-  sharing is compartmentalized into groups of four input  channels and then groups of three input channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  For better representation, a quaternion convolution  layer is also used in the stem layer (first layer of the net-  work) as a quaternion-based frontend layer and the fully-  connected dense layer as a PHM-based backend layer of  deep axial-hypercomplex networks (DANs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Figure 1  illustrates our proposed axial-hypercomplex network ar-  chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Experiment  We conduct an extensive experiment on four classi-  fication datasets to analyze the effectiveness of our pro-  posed axial-hypercomplex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' As QCNNs, VCNNs,  residual networks (ResNets), QPHM [16], and VPHM  [16] all are performed 2D spatial convolution operation,  Model Name  Layers  Dataset  Params  FLOPS  Latency  Validation  Accuracy  ResNet [6]  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content='75  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Image classification performance on the CIFAR benchmarks for 26, 35, and 50-layer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Here, QPHM, and  VPHM define the quaternion networks with PHM FC layer, and vectormap networks with the PHM FC layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  therefore we compare our proposed axial hypercomplex  networks performance with the above-mentioned base-  line models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Among them, all models perform Hamilto-  nian products like our proposed model except ResNets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Method  We conducted our experiments by using five-  dimensional PHM dense layer in the backend of the  network, quaternion network at the beginning of the  Model Name  Layers  Params  FLOPS  Latency  Validation Accuracy  ResNet [6]  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content='05  ResNet [6]  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content='90  ResNet [6]  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content=' Image classification performance on the SVHN benchmarks for 26, 35, and 50-layer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Here, QPHM, and  VPHM define the quaternion networks with PHM FC layer, and vectormap networks with PHM FC layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Model Name  Layers  Params  FLOPS  Latency  Validation Accuracy  ResNet [6]  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content='06  ResNet [6]  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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+page_content='42  VPHM [16]  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='7M  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='75G  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='51ms  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='96  Axial-Hypercomplex  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='7M  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='75G  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='93ms  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='73-62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='07  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Image classification performance on the Tiny ImageNet benchmarks for 26, 35, and 50-layer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Here, QPHM,  and VPHM define the quaternion networks with PHM FC layer, and vectormap networks with PHM FC layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  network, and axial-hypercomplex residual bottleneck  block on CIFAR benchmark datasets [10], Street View  House Numbers (SVHN) [12], and Tiny ImageNet [11]  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  The models we tested to compare with our proposed  model, are: the standard DCNNs [6], the DQNNs [4],  the axial-ResNet with QPHM [16], QPHM [16], VPHM  [16], and our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' CIFAR-10 and CIFAR-  100 datasets consist of 60,000 color images of size 32  × 32 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These datasets fall into 10 and 100 dis-  tinct classes and are split into a training set with 50,000  images and a test set with 10,000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' We perform  standard data augmentation schemes for these datasets  like [4–6,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Both datasets were normalized using per-  channel mean and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' We perform hori-  zontal flips and take random crops from images padded  by 4 pixels on each side to obtain a 40 × 40 pixel image,  then a 32 × 32 crop is randomly extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  SVHN contains about 600,000 digit images [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' For  experiments on SVHN we don’t do any image pre-  processing, except simple mean/std normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' We  use similar augmentation for the Tiny ImageNet dataset  which contains 100,000 training images of 200 classes  (500 for each class) downsized to 64×64 colored images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  The test set contains 10,000 images [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  All baseline models were trained using the same  components as the real-valued networks, the original  quaternion network, the original vectormap network,  the QPHM, and the VPHM networks using the same  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  All models in Table 2 were trained using  the same hyperparameters and the same number of out-  put channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The 50-layer architectural details of the  above-mentioned models are depicted in Table 1 for the  CIFAR-100 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Due to space limitation, the deep  ResNets and VPHM network architectures are not de-  picted in the architecture Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  In the stem layer, the 3 × 3 convolution network  is used for deep ResNets [6], 3 × 3 quaternion net-  work is used for the deep quaternion ResNets [4, 18],  for the QPHM [16], and axial-hypercomplex networks  (our proposed method), and 3 × 3 vectormap network is  used for the deep vectormap ResNets [5], and the VPHM  [16] networks with stride 1 & 120 output filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' We  use parameterized hypercomplex multiplication (PHM)  for the dense layer in the backend of deep ResNets,  QPHM, VPHM, and our proposed axial-hypercomplex  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' In the bottleneck block, the number of out-  put channels of bottleneck groups are 120, 240, 480, &  960 for all networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' In this experiment, we analyze 26-  layer, 35-layer, and 50-layer architectures with the bot-  tleneck block multipliers “[1, 2, 4, 1]”, “[2, 3, 4, 2]”, and  “[3, 4, 6, 3]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These are depicted in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  We ran all of the models using stochastic gradient de-  cent optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' We used linearly warmed-up learning  from zero to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='1 for the first 10 epochs and then used  cosine learning rate scheduling from epochs 11 to 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  All models were trained for 128 batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Results  The overall results of all models (base models and  our proposed networks) appear in Tables 2, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  The top half of Table 2 shows the results for the CI-  FAR10 dataset and the bottom half presents the results  for the CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Both datasets have been tested by  the 26, 35, and 50 layers architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' These are the pa-  rameter count, FLOPS count (number of multiply-add  operations), inference time or Latency (time required  to process a single image), and the percentage accu-  racy of validation results for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' We evaluate  original ResNets [6], ResNet with QPHM [16], orig-  inal quaternion networks [4], original vectormap net-  works [5], QPHM [16], and VPHM [16] with the same  configuration like our proposed axial-hypercomplex net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Our proposed axial-hypercomplex networks per-  form better in validation accuracy with lower param-  eter count and FLOPS for CIFAR-10 and CIFAR-100  datasets than the baseline networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  More precisely,  our proposed method takes almost 6 times, 1/3 times,  1/2 times, 1/3 times, and 1/2 times fewer parameters  than the ResNets, quaternion networks, vectormap net-  works, QPHM, and VPHM respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Moreover,  axial-hypercomplex networks achieved state-of-the-art  results for these CIFAR benchmarks in hypercomplex  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  The performances for SVHN and Tiny ImageNet  datasets are shown in Tables 3 and 4 for all architec-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' The axial-hypercomplex network’s validation ac-  curacies outperform the other base networks with fewer  trainable parameters and FLOPS like CIFAR datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  The result Tables 2, 3, and 4 show our proposed model  performance ranges of three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' However, the latency  of axial-hypercomplex networks is a little bit higher in  some cases than the quaternion-based networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' This  may be due to the use of vectormap networks along with  quaternion networks as the latency for vectormap net-  works is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Discussion and Conclusions  This paper proposes axial-hypercomplex convolu-  tions to reduce the cost of 2D convolutional opera-  tions and shows the effectiveness of image classifi-  cation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  We also applied 4D PHM in the net-  work’s backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  On CIFAR benchmarks, our pro-  posed Axial-hypercomplex network, formed by stacking  axial-vectormap convolution (three-dimensional) in the  quaternion bottleneck blocks, achieved state-of-the-art  results among hypercomplex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Our main conclusion is that using quaternion convo-  lutions as the frontend stem layer, four/five-dimensional  PHM-based densely connected backend layer, and axial-  hypercomplex bottleneck block improves classification  performance on the CIFAR benchmarks, SVHN, and  Tiny ImageNet datasets in comparison to the other  models we tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Our proposed method factorizes a  channel-wise 2D convolution (hypercomplex convolu-  tion which works along the channels) to a column con-  volution and a row convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Extensive experiments  show that this leads to systematic improvement with far  fewer trainable parameters on image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' This  proposed method can save 33%, and 50% trainable pa-  rameters compared to original quaternion and vectormap  networks and QPHM and VPHM networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Although our proposed axial-hypercomplex design  reduced parameter counts and FLOPS, it exhibited  higher latency than real-valued and hypercomplex-  valued convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  This is because the  model performs convolution twice (height-axis and  width-axis) and it takes transition time from 2D convo-  lution to two consecutive 1D convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' As we re-  placed spatial quaternion (four-dimensional hypercom-  plex network) 2D convolution using two axial vec-  tormap (three-dimensional hypercomplex network) 1D  convolutions, the number of output channels are re-  stricted to 120 or a multiple of 120 which are divisible by  three and four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Our investigation concludes that the per-  formance comparison between the hypercomplex net-  works and our proposed axial-hypercomplex networks  shows that the axial-hypercomplex convolution provides  better validation performance with fewer trainable pa-  rameters and FLOPS for image classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Further work may be directed toward the architecture  of the axial quaternion network and axial vectormap net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content=' Moreover, other datasets will be tested to check  whether these proposed architectures can perform in a  similar manner or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
+page_content='  Finally, axial-quaternion and  axial-vectormap convolutional methods will help to re-  move the number of output channels constrained as it  will divisible by four for axial-quaternion networks and  three for axial-vectormap networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E3T4oBgHgl3EQfngq-/content/2301.04626v1.pdf'}
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