LieBN as a Natural Generalization of Euclidean BN, And Backpropagation of Matrix Functions

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

D LIEBN AS A NATURAL GENERALIZATION OF EUCLIDEAN BN

The centering and biasing in Euclidean BN correspond to the group action of R. From a geometric perspective, the standard Euclidean metric is invariant under this group operation. Consequently, it is not surprising that our LieBN algorithm formulated in Alg. 1 serves as a natural generalization of standard Euclidean batch normalization. We formalize this fact in the following proposition.

E DOMAIN-SPECIFIC MOMENTUM LIEBN FOR EEG CLASSIFICATION

where i is the index of the domain. We follow the official code of SPDDSMBN[5] to implement our DSMLieBN. In a word, the only difference between DSMLieBN and SPDDSMBN is the different way of normalization.

Analogous to Thm. 5.3, computations for DSMLieBN under pullback metrics can also be performed by mapping, calculating, and then remapping.

[5] https://github.com/rkobler/TSMNet

F BACKPROPAGATION OF MATRIX FUNCTIONS

Our implementation of LieBN on SPD manifolds involves several matrix functions. Thus, we employ matrix backpropagation (BP) (Ionescu et al., 2015) for gradient computation. These matrix operations can be divided into Cholesky decomposition and the functions based on Eigende-composition.

The differentiation of the Cholesky decomposition can be found in Murray (2016, Eq. 8) or Lin (2019, Props. 4). Besides, our homemade BP of the Cholesky decomposition yields a similar gradient to the one generated by autograd of torch.linalg.cholesky. Therefore, during the experiments, we use torch.linalg.cholesky.

where ∇XL is the Euclidean gradient of the loss function L w.r.t. X. Matrix K is defined as