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 2 PRELIMINARIES This section provides a brief review of the Lie group and the geometries of SPD manifolds. For more in-depth discussions, please refer to Tu (2011); Do Carmo & Flaherty Francis (1992). A Lie group is a group and also a manifold. The most natural Riemannian metric on a Lie group is the left-invariant metric[1]. Similarly, one can define the right-invariant metric as Def. 2.2. A biinvariant Riemannian metric is the one with both left and right invariance. Given the analogous properties of left and right-invariant metrics, this paper focuses on left-invariant metrics. The idea of pullback is ubiquitous in differential geometry and can be considered as a natural counterpart of bijection in the set theory. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. [1] Left invariant metric always exists for every Lie group (Do Carmo & Flaherty Francis, 1992). Authors:
(1) Ziheng Chen, University of Trento;
(2) Yue Song, University of Trento and a Corresponding author;
(3) Yunmei Liu, University of Louisville;
(4) Nicu Sebe, University of Trento. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Preliminaries 2 Preliminaries 3. Revisiting Normalization 3.1 Revisiting Euclidean Normalization 3.1 Revisiting Euclidean Normalization 3.2 Revisiting Existing RBN 3.2 Revisiting Existing RBN 4 Riemannian Normalization on Lie Groups 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 LieBN on the Lie Groups of SPD Manifolds and 5.1 Deformed Lie Groups of SPD Manifolds 5.2 LieBN on SPD Manifolds 5.2 LieBN on SPD Manifolds 6 Experiments 6 Experiments 6.1 Experimental Results 6.1 Experimental Results 7 Conclusions, Acknowledgments, and References 7 Conclusions, Acknowledgments, and References APPENDIX CONTENTS APPENDIX CONTENTS A Notations A Notations B Basic layes in SPDnet and TSMNet B Basic layes in SPDnet and TSMNet C Statistical Results of Scaling in the LieBN C Statistical Results of Scaling in the LieBN D LieBN as a Natural Generalization of Euclidean BN D LieBN as a Natural Generalization of Euclidean BN E Domain-specific Momentum LieBN for EEG Classification E Domain-specific Momentum LieBN for EEG Classification F Backpropagation of Matrix Functions F Backpropagation of Matrix Functions G Additional Details and Experiments of LieBN on SPD manifolds G Additional Details and Experiments of LieBN on SPD manifolds H Preliminary Experiments on Rotation Matrices 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 2 PRELIMINARIES This section provides a brief review of the Lie group and the geometries of SPD manifolds. For more in-depth discussions, please refer to Tu (2011); Do Carmo & Flaherty Francis (1992). A Lie group is a group and also a manifold. The most natural Riemannian metric on a Lie group is the left-invariant metric[1]. Similarly, one can define the right-invariant metric as Def. 2.2. A biinvariant Riemannian metric is the one with both left and right invariance. Given the analogous properties of left and right-invariant metrics, this paper focuses on left-invariant metrics. The idea of pullback is ubiquitous in differential geometry and can be considered as a natural counterpart of bijection in the set theory. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. available on arxiv [1] Left invariant metric always exists for every Lie group (Do Carmo & Flaherty Francis, 1992). Authors: (1) Ziheng Chen, University of Trento; (2) Yue Song, University of Trento and a Corresponding author; (3) Yunmei Liu, University of Louisville; (4) Nicu Sebe, University of Trento. Authors: Authors: (1) Ziheng Chen, University of Trento; (2) Yue Song, University of Trento and a Corresponding author; (3) Yunmei Liu, University of Louisville; (4) Nicu Sebe, University of Trento.

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A Brief Review of the Lie Group and the Geometries of SPD Manifolds

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