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Riemannian Multinomial Logistics Regression for SPD Neural Networks

Ziheng Chen, Yue Song, Gaowen Liu, Ramana Rao Kompella, Xiaojun Wu, Nicu Sebe

TL;DR

This work introduces a unified framework for building Riemannian classifiers under the metrics pulled back from the Euclidean space, and showcases the framework under the parameterized Log-Euclidean Metric (LEM) and Log-Cholesky Metric (LCM).

Abstract

Deep neural networks for learning Symmetric Positive Definite (SPD) matrices are gaining increasing attention in machine learning. Despite the significant progress, most existing SPD networks use traditional Euclidean classifiers on an approximated space rather than intrinsic classifiers that accurately capture the geometry of SPD manifolds. Inspired by Hyperbolic Neural Networks (HNNs), we propose Riemannian Multinomial Logistics Regression (RMLR) for the classification layers in SPD networks. We introduce a unified framework for building Riemannian classifiers under the metrics pulled back from the Euclidean space, and showcase our framework under the parameterized Log-Euclidean Metric (LEM) and Log-Cholesky Metric (LCM). Besides, our framework offers a novel intrinsic explanation for the most popular LogEig classifier in existing SPD networks. The effectiveness of our method is demonstrated in three applications: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/SPDMLR.git.

Riemannian Multinomial Logistics Regression for SPD Neural Networks

TL;DR

This work introduces a unified framework for building Riemannian classifiers under the metrics pulled back from the Euclidean space, and showcases the framework under the parameterized Log-Euclidean Metric (LEM) and Log-Cholesky Metric (LCM).

Abstract

Deep neural networks for learning Symmetric Positive Definite (SPD) matrices are gaining increasing attention in machine learning. Despite the significant progress, most existing SPD networks use traditional Euclidean classifiers on an approximated space rather than intrinsic classifiers that accurately capture the geometry of SPD manifolds. Inspired by Hyperbolic Neural Networks (HNNs), we propose Riemannian Multinomial Logistics Regression (RMLR) for the classification layers in SPD networks. We introduce a unified framework for building Riemannian classifiers under the metrics pulled back from the Euclidean space, and showcase our framework under the parameterized Log-Euclidean Metric (LEM) and Log-Cholesky Metric (LCM). Besides, our framework offers a novel intrinsic explanation for the most popular LogEig classifier in existing SPD networks. The effectiveness of our method is demonstrated in three applications: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/SPDMLR.git.
Paper Structure (27 sections, 10 theorems, 43 equations, 1 figure, 7 tables)

This paper contains 27 sections, 10 theorems, 43 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemannian geometry
  4. The geometry of SPD manifolds
  5. SPD MLRs on SPD manifolds
  6. Reformulation of Euclidean MLR
  7. SPD MLRs under PEMs
  8. SPD MLRs under deformed LEM and LCM
  9. Rethinking the existing LogEig classifier
  10. Experiments
  11. Baseline models
  12. Datasets
  13. Implementation details
  14. Experimental results
  15. Model efficiency
  16. ...and 12 more sections

Key Result

Theorem 2.2

Let $S, S_1,S_2 \in \mathcal{S}^{n}_{++}$ and $V_1, V_2 \in T_S\mathcal{S}^{n}_{++}$, $\phi:\mathcal{S}^{n}_{++} \rightarrow \mathcal{S}^{n}$ is a diffeomorphism. We define the following operations, where $\phi_{*,S}: T_S\mathcal{S}^{n}_{++} \rightarrow T_{\phi(S)}\mathcal{S}^{n}$ is the differential map of $\phi$ at $S$, and $\langle \cdot,\cdot \rangle$ is the standard Frobenius inner product.

Figures (1)

  • Figure 1: Conceptual illustration of SPD hyperplanes induced by $(\alpha,\beta)\text{-LEM}$ and $(\theta)\text{-LCM}$. In each subfigure, the black dots are symmetric positive semi-definite (SPSD) matrices, denoting the boundary of $\mathcal{S}^{2}_{++}$, while the blue, red, and yellow dots denote three SPD hyperplanes.

Theorems & Definitions (35)

  • Definition 2.1: Pullback Metrics
  • Theorem 2.2: Pullback Euclidean Metrics (PEMs)
  • Definition 3.1: SPD hyperplanes
  • Definition 3.2: SPD MLR
  • Proposition 3.3: Submanifolds
  • proof
  • Remark 3.4: Difference with the gyro SPD MLR
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 25 more