Table of Links

Abstract and 1. Introduction

2. Preliminaries

3. 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

4. Experiments

5. 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.

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.