LieBN on SPD Manifolds: Showcasing Our LieBN Framework

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

5.2 LIEBN ON SPD MANIFOLDS

Now, we showcase our LieBN framework illustrated in Alg. 1 on SPD manifolds. As discussed in Sec. 5.1, there are three families of left-invariant metrics, namely (θ, α, β)-AIM, (α, β)-LEM, and θ-LCM. Since all three metric families are pullback metrics, the LieBN based on these metrics can be simplified and calculated in the co-domain. We denote Alg. 1 as

Then we can obtain the following theorem.