Proofs of the Lemmas and Theories in the Main Paper

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

I PROOFS OF THE LEMMAS AND THEORIES IN THE MAIN PAPER

Proof of Prop. 4.1. Property 1:

The MLE of M is

Property 2:

The first equation is obtained by (Pennec, 2004, Thm. 7), while the last equation is obtained by the isometry of the left translation.

By Eq. (49), we can readily obtain the results.

Then we have

we can further simply the above equation as