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Matrix Manifold Neural Networks++

Xuan Son Nguyen, Shuo Yang, Aymeric Histace

TL;DR

This work advances neural networks on SPD and Grassmann manifolds by embedding gyrovector-space principles into practical building blocks. It generalizes fully-connected and convolutional layers to SPD manifolds via SPD hypergyroplanes and gyrodistances, develops MLR on SPSD structure spaces, and enables backpropagation for Grassmann networks using a projector-perspective Grassmann log map. The approach also extends to graph-structured data with Grassmann-based GCNs, and is validated on human action recognition and node classification, showing improvements over strong manifold-based baselines. Overall, the paper provides a cohesive, mathematically grounded framework for broadening deep learning to matrix and subspace manifolds with concrete architectures and training strategies.

Abstract

Deep neural networks (DNNs) on Riemannian manifolds have garnered increasing interest in various applied areas. For instance, DNNs on spherical and hyperbolic manifolds have been designed to solve a wide range of computer vision and nature language processing tasks. One of the key factors that contribute to the success of these networks is that spherical and hyperbolic manifolds have the rich algebraic structures of gyrogroups and gyrovector spaces. This enables principled and effective generalizations of the most successful DNNs to these manifolds. Recently, some works have shown that many concepts in the theory of gyrogroups and gyrovector spaces can also be generalized to matrix manifolds such as Symmetric Positive Definite (SPD) and Grassmann manifolds. As a result, some building blocks for SPD and Grassmann neural networks, e.g., isometric models and multinomial logistic regression (MLR) can be derived in a way that is fully analogous to their spherical and hyperbolic counterparts. Building upon these works, we design fully-connected (FC) and convolutional layers for SPD neural networks. We also develop MLR on Symmetric Positive Semi-definite (SPSD) manifolds, and propose a method for performing backpropagation with the Grassmann logarithmic map in the projector perspective. We demonstrate the effectiveness of the proposed approach in the human action recognition and node classification tasks.

Matrix Manifold Neural Networks++

TL;DR

This work advances neural networks on SPD and Grassmann manifolds by embedding gyrovector-space principles into practical building blocks. It generalizes fully-connected and convolutional layers to SPD manifolds via SPD hypergyroplanes and gyrodistances, develops MLR on SPSD structure spaces, and enables backpropagation for Grassmann networks using a projector-perspective Grassmann log map. The approach also extends to graph-structured data with Grassmann-based GCNs, and is validated on human action recognition and node classification, showing improvements over strong manifold-based baselines. Overall, the paper provides a cohesive, mathematically grounded framework for broadening deep learning to matrix and subspace manifolds with concrete architectures and training strategies.

Abstract

Deep neural networks (DNNs) on Riemannian manifolds have garnered increasing interest in various applied areas. For instance, DNNs on spherical and hyperbolic manifolds have been designed to solve a wide range of computer vision and nature language processing tasks. One of the key factors that contribute to the success of these networks is that spherical and hyperbolic manifolds have the rich algebraic structures of gyrogroups and gyrovector spaces. This enables principled and effective generalizations of the most successful DNNs to these manifolds. Recently, some works have shown that many concepts in the theory of gyrogroups and gyrovector spaces can also be generalized to matrix manifolds such as Symmetric Positive Definite (SPD) and Grassmann manifolds. As a result, some building blocks for SPD and Grassmann neural networks, e.g., isometric models and multinomial logistic regression (MLR) can be derived in a way that is fully analogous to their spherical and hyperbolic counterparts. Building upon these works, we design fully-connected (FC) and convolutional layers for SPD neural networks. We also develop MLR on Symmetric Positive Semi-definite (SPSD) manifolds, and propose a method for performing backpropagation with the Grassmann logarithmic map in the projector perspective. We demonstrate the effectiveness of the proposed approach in the human action recognition and node classification tasks.
Paper Structure (83 sections, 9 theorems, 122 equations, 2 figures, 14 tables, 1 algorithm)

This paper contains 83 sections, 9 theorems, 122 equations, 2 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SPD Manifolds
  4. Grassmann Manifolds
  5. Neural Networks on SPD and Grassmann Manifolds
  6. Neural Networks on SPD Manifolds
  7. Neural Networks on Grassmann Manifolds
  8. Proposed Approach
  9. Notation
  10. Neural Networks on SPD Manifolds
  11. FC Layers in SPD Neural Networks
  12. Proof
  13. Proof
  14. Proof
  15. Proof
  16. ...and 68 more sections

Key Result

Proposition 3.2

Let $\mathbf{P} \in \operatorname{Sym}_n^{+,g}$, $\mathbf{W} \in T_{\mathbf{P}} \operatorname{Sym}_n^{+,g}$, and $\mathcal{H}^{spd,g}_{\mathbf{W},\mathbf{P}}$ be the SPD hypergyroplanes defined in Definition def:spd_hypergyroplanes. Then where $\mathbf{I}_n$ denotes the $n \times n$ identity matrix, and $\langle .,. \rangle^g$ is the SPD inner product in $\operatorname{Sym}_n^{+,g}$NguyenGyroMatM

Figures (2)

  • Figure 1: The pipelines of GyroSpd++ (left) and Gr-GCN++ (right).
  • Figure 2: Computation of Pseudo-gyrodistances

Theorems & Definitions (30)

  • Definition 3.1: SPD Hypergyroplanes NguyenGyroMatMans23
  • Proposition 3.2
  • Definition 3.3
  • Proposition 3.4: FC layers with Affine-Invariant Metrics
  • Proposition 3.5: FC layers with Log-Euclidean Metrics
  • Proposition 3.6: FC layers with Log-Cholesky Metrics
  • Definition 3.7: The Binary Operation in Structure Spaces
  • Definition 3.8: The Inverse Operation in Structure Spaces
  • Definition 3.9: The Inner Product in Structure Spaces
  • Definition 3.10: Hypergyroplanes in Structure Spaces
  • ...and 20 more