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 A NOTATIONS For better clarity, we summarize all the notations used in this paper in Tab. 6. B BASIC LAYERS IN SPDNET AND TSMNET SPDNet (Huang & Van Gool, 2017) is the most classic SPD neural network. SPDNet mimics the conventional densely connected feedforward network, consisting of three basic building blocks where max(·) is element-wise maximization. BiMap and ReEig mimic transformation and nonlinear activation, while LogEig maps SPD matrices into the tangent space at the identity matrix for classification. C STATISTICAL RESULTS OF SCALING IN THE LIEBN In this section, we will show the effect of our scaling (Eq. (14)) on the population. We will see that while the resulting population variance is generally agnostic, it becomes analytic under certain circumstances, such as SPD manifolds under LEM or LCM. As a result, Eq. (14) can normalize and transform the latent Gaussian distribution. The above lemma implies that when ∆ is a constant, Y also follows a Gaussian distribution. By Prop. C.3, we can directly obtain the following corollary. This paper is available on arxiv under CC BY-NC-SA 4.0 DEED license. 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 A NOTATIONS For better clarity, we summarize all the notations used in this paper in Tab. 6. B BASIC LAYERS IN SPDNET AND TSMNET SPDNet (Huang & Van Gool, 2017) is the most classic SPD neural network. SPDNet mimics the conventional densely connected feedforward network, consisting of three basic building blocks where max(·) is element-wise maximization. BiMap and ReEig mimic transformation and nonlinear activation, while LogEig maps SPD matrices into the tangent space at the identity matrix for classification. C STATISTICAL RESULTS OF SCALING IN THE LIEBN In this section, we will show the effect of our scaling (Eq. (14)) on the population. We will see that while the resulting population variance is generally agnostic, it becomes analytic under certain circumstances, such as SPD manifolds under LEM or LCM. As a result, Eq. (14) can normalize and transform the latent Gaussian distribution. The above lemma implies that when ∆ is a constant, Y also follows a Gaussian distribution. By Prop. C.3, we can directly obtain the following corollary. 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 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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Basic Layers In SPDNET, TSMNET, and Statistical Results of Scaling in the LieBN

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