Preliminary Experiments on Rotation Matrices

Table of Links

Abstract and 1 Introduction

2 Preliminaries

3. Revisiting Normalization

3.1 Revisiting Euclidean Normalization

3.2 Revisiting Existing RBN

4 Riemannian Normalization on Lie Groups

5 LieBN on the Lie Groups of SPD Manifolds and 5.1 Deformed Lie Groups of SPD Manifolds

5.2 LieBN on SPD Manifolds

6 Experiments

6.1 Experimental Results

7 Conclusions, Acknowledgments, and References

APPENDIX CONTENTS

A Notations

B Basic layes in SPDnet and TSMNet

C Statistical Results of Scaling in the LieBN

D LieBN as a Natural Generalization of Euclidean BN

E Domain-specific Momentum LieBN for EEG Classification

F Backpropagation of Matrix Functions

G Additional Details and Experiments of LieBN on SPD manifolds

H Preliminary Experiments on Rotation Matrices

I Proofs of the Lemmas and Theories in the Main Paper

H PRELIMINARY EXPERIMENTS ON ROTATION MATRICES

This section implements our LieBN in Alg. 1 on the special orthogonal groups, i.e.,SO(n), also known as rotation matrices. We apply our LieBN to the classic LieNet (Huang & Van Gool, 2017), where the latent space is the special orthogonal group.

H.1 GEOMETRY ON ROTATION MATRICES

We denote R, S ∈ SO(n), and γ(R,S)(t) as the geodesic connecting R and S. The neutral elements of rotation matrices is the identity matrix. Tab. 8 summarizes all the necessary Riemannian ingredients of the invariant metric on SO(n)

For the specific SO(3), the matrix logarithm and exponentiation can be calculated without decomposition (Murray et al., 2017, Exs. A. 11 and A.14)

H.2 DATASETS AND PREPROCESSING

Following LieNet, we validate our LieBN on the G3D dataset (Bloom et al., 2012). This dataset (Bloom et al., 2012) consists of 663 sequences of 20 different gaming actions. Each sequence is recorded by 3D locations of 20 joints (i.e., 19 bones). Following Huang & Van Gool (2017), we use the code of Vemulapalli et al. (2014) to represent each skeleton sequence as a point on the Lie group SON×T (3), where N and T denote spatial and temporal dimensions. As preprocessed in Huang & Van Gool (2017), we set T as 100 for each sequence on the G3D.

H.3 IMPLEMENTATION DETAILS

LieNet: The LieNet consists of three basic layers: RotMap, RotPooling, and LogMap layers. The RotMap mimics the convolutional layer, while the RotPooling extends the pooling layers to rotation matrices. The logMap layer maps the rotation matrix into the tangent space at the identity for classification. Note that the official code of LieNet[8] is developed by Matlab. We follow the opensourced Pytorch code[9] to implement our experiments. To reproduce LieNet more faithfully, we made the following modifications to this Pytorch code. We re-code the LogMap and RotPooling layers to make them consistent with the official Matlab implementation. In addition, we also extend the existing Riemannian optimization package geoopt B´ecigneul & Ganea (2018) into SO(3) to allow for Riemannian version of SGD, ADAM, and AMSGrad on SO(3), which is missing in the current package. However, we find that SGD is the best optimizer for LieNet. Therefore, we adopt SGD during the experiments. We apply our LieBN before the LogMap layer and refer to this network as LieNetLieBN. Note that the dimension of features in LieNet is B × N ×T ×3×3, we calculate Lie group statistics along the batch and spatial dimensions (B × T ), resulting in an N × 3 × 3 running mean.

H.4 RESULTS

The 10-fold results are shown in Tab. 9. Due to different software, our reimplemented LieNet is slightly worse than the performance reported in Huang et al. (2017). However, we still can observe a clear improvement of LieNetLieBN over LieNet.

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[8] https://github.com/zhiwu-huang/LieNet

[9] https://github.com/hjf1997/LieNet