Table of Links Abstract and 1. Introduction Preliminaries


Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds


Experiments


Conclusion and References A. Notations B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition E. Node Classification F. Limitations of our work G. Some Related Definitions H. Computation of Canonical Representation I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 4 EXPERIMENTS 4.1 HUMAN ACTION RECOGNITION We use three datasets, i.e., HDM05 (Muller et al., 2007), FPHA (Garcia-Hernando et al., 2018), and ¨ NTU RBG+D 60 (NTU60) (Shahroudy et al., 2016). We compare our networks against the following state-of-the-art models: SPDNet (Huang & Gool, 2017)[1], SPDNetBN (Brooks et al., 2019)[2], SPSDAI (Nguyen, 2022a), GyroAI-HAUNet (Nguyen, 2022b), and MLR-AI (Nguyen & Yang, 2023). 4.1.1 ABLATION STUDY Convolutional layers in SPD neural networks Our network GyroSpd++ has a MLR layer stacked on top of a convolutional layer (see Fig. 1). The motivation for using a convolutional layer is that it can extract global features from local ones (covariance matrices computed from joint coordinates within sub-sequences of an action sequence). We use Affine-Invariant metrics for the convolutional layer and Log-Euclidean metrics for the MLR layer. Results in Tab. 1 show that GyroSpd++ consistently outperforms the SPD baselines in terms of mean accuracy. Results of GyroSpd++ with different designs of Riemannian metrics for its layers are given in Appendix D.4.1. MLR in structure spaces We build GyroSpsd++ by replacing the MLR layer of GyroSpd++ with a MLR layer proposed in Section 3.3. Results of GyroSpsd++ are given in Tab. 1. Except SPSDAI, GyroSpsd++ outperforms the other baselines on HDM05 dataset in terms of mean accuracy. Furthermore, GyroSpsd++ outperforms GyroSpd++ and all the baselines on FPHA and NTU60 datasets in terms of mean accuracy. These results show that MLR is effective when being designed in structure spaces from a gyrovector space perspective. 4.2 NODE CLASSIFICATION We use three datasets, i.e., Airport (Zhang & Chen, 2018), Pubmed (Namata et al., 2012a), and Cora (Sen et al., 2008), each of them contains a single graph with thousands of labeled nodes. We compare our network Gr-GCN++ (see Fig. 1) against its variant Gr-GCN-ONB (see Appendix E.2.4) based on the ONB perspective. Results are shown in Tab. 2. Both networks give the best performance for n = 14 and p = 7. It can be seen that Gr-GCN++ outperforms Gr-GCN-ONB in all cases. The performance gaps are significant on Pubmed and Cora datasets. Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. [1] https://github.com/zhiwu-huang/SPDNet. [2] https://papers.nips.cc/paper/2019/hash/6e69ebbfad976d4637bb4b39de261bf7-Abstract. html. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries Proposed Approach
3.1 Notation
3.2 Nueral Networks on SPD Manifolds
3.3 MLR in Structure Spaces
3.4 Neural Networks on Grassmann Manifolds Experiments Conclusion and References Preliminaries Preliminaries Preliminaries Proposed Approach 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 N. Proof of Proposition 3.12 4 EXPERIMENTS 4.1 HUMAN ACTION RECOGNITION We use three datasets, i.e., HDM05 (Muller et al., 2007), FPHA (Garcia-Hernando et al., 2018), and ¨ NTU RBG+D 60 (NTU60) (Shahroudy et al., 2016). We compare our networks against the following state-of-the-art models: SPDNet (Huang & Gool, 2017)[1], SPDNetBN (Brooks et al., 2019)[2], SPSDAI (Nguyen, 2022a), GyroAI-HAUNet (Nguyen, 2022b), and MLR-AI (Nguyen & Yang, 2023). 4.1.1 ABLATION STUDY 4.1.1 ABLATION STUDY Convolutional layers in SPD neural networks Our network GyroSpd++ has a MLR layer stacked on top of a convolutional layer (see Fig. 1). The motivation for using a convolutional layer Convolutional layers in SPD neural networks is that it can extract global features from local ones (covariance matrices computed from joint coordinates within sub-sequences of an action sequence). We use Affine-Invariant metrics for the convolutional layer and Log-Euclidean metrics for the MLR layer. Results in Tab. 1 show that GyroSpd++ consistently outperforms the SPD baselines in terms of mean accuracy. Results of GyroSpd++ with different designs of Riemannian metrics for its layers are given in Appendix D.4.1. MLR in structure spaces We build GyroSpsd++ by replacing the MLR layer of GyroSpd++ with a MLR layer proposed in Section 3.3. Results of GyroSpsd++ are given in Tab. 1. Except SPSDAI, GyroSpsd++ outperforms the other baselines on HDM05 dataset in terms of mean accuracy. Furthermore, GyroSpsd++ outperforms GyroSpd++ and all the baselines on FPHA and NTU60 datasets in terms of mean accuracy. These results show that MLR is effective when being designed in structure spaces from a gyrovector space perspective. MLR in structure spaces 4.2 NODE CLASSIFICATION We use three datasets, i.e., Airport (Zhang & Chen, 2018), Pubmed (Namata et al., 2012a), and Cora (Sen et al., 2008), each of them contains a single graph with thousands of labeled nodes. We compare our network Gr-GCN++ (see Fig. 1) against its variant Gr-GCN-ONB (see Appendix E.2.4) based on the ONB perspective. Results are shown in Tab. 2. Both networks give the best performance for n = 14 and p = 7. It can be seen that Gr-GCN++ outperforms Gr-GCN-ONB in all cases. The performance gaps are significant on Pubmed and Cora datasets. Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv [1] https://github.com/zhiwu-huang/SPDNet. [2] https://papers.nips.cc/paper/2019/hash/6e69ebbfad976d4637bb4b39de261bf7-Abstract. html.

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Bridging Geometry and Deep Learning: Key Developments in SPD and Grassmann Networks

Graph Embeddings and Node Learning on Grassmann Manifolds

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New Riemannian Networks Outperform Traditional Models in Action Recognition and Node Classification

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