Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Siegel Neural Networks

Xuan Son Nguyen, Aymeric Histace, Nistor Grozavu

TL;DR

This work develops discriminative neural networks on Siegel spaces, a class of Riemannian symmetric spaces (RSS) that extend hyperbolic and SPD geometries. It introduces MLR layers based on the quotient structure $ ext{SH}_m \\cong Sp_{2m}/SpO_{2m}$ and a vector-valued distance $d_{\\Delta}(\cdot,\cdot)$, along with two FC-layer families (AFC and DFC) leveraging the real symplectic group action on $\\text{SH}_m$. A closed-form point-to-hyperplane distance on Siegel spaces is derived, with extensions to product spaces to enable compact, invariant decision boundaries for multiclass problems. Empirically, the approach achieves state-of-the-art performance on radar clutter classification and node classification, underscoring the practical value of non-Euclidean representations for discriminative learning on RSS.

Abstract

Riemannian symmetric spaces (RSS) such as hyperbolic spaces and symmetric positive definite (SPD) manifolds have become popular spaces for representation learning. In this paper, we propose a novel approach for building discriminative neural networks on Siegel spaces, a family of RSS that is largely unexplored in machine learning tasks. For classification applications, one focus of recent works is the construction of multiclass logistic regression (MLR) and fully-connected (FC) layers for hyperbolic and SPD neural networks. Here we show how to build such layers for Siegel neural networks. Our approach relies on the quotient structure of those spaces and the notation of vector-valued distance on RSS. We demonstrate the relevance of our approach on two applications, i.e., radar clutter classification and node classification. Our results successfully demonstrate state-of-the-art performance across all datasets.

Siegel Neural Networks

TL;DR

This work develops discriminative neural networks on Siegel spaces, a class of Riemannian symmetric spaces (RSS) that extend hyperbolic and SPD geometries. It introduces MLR layers based on the quotient structure and a vector-valued distance , along with two FC-layer families (AFC and DFC) leveraging the real symplectic group action on . A closed-form point-to-hyperplane distance on Siegel spaces is derived, with extensions to product spaces to enable compact, invariant decision boundaries for multiclass problems. Empirically, the approach achieves state-of-the-art performance on radar clutter classification and node classification, underscoring the practical value of non-Euclidean representations for discriminative learning on RSS.

Abstract

Riemannian symmetric spaces (RSS) such as hyperbolic spaces and symmetric positive definite (SPD) manifolds have become popular spaces for representation learning. In this paper, we propose a novel approach for building discriminative neural networks on Siegel spaces, a family of RSS that is largely unexplored in machine learning tasks. For classification applications, one focus of recent works is the construction of multiclass logistic regression (MLR) and fully-connected (FC) layers for hyperbolic and SPD neural networks. Here we show how to build such layers for Siegel neural networks. Our approach relies on the quotient structure of those spaces and the notation of vector-valued distance on RSS. We demonstrate the relevance of our approach on two applications, i.e., radar clutter classification and node classification. Our results successfully demonstrate state-of-the-art performance across all datasets.

Paper Structure

This paper contains 25 sections, 4 theorems, 37 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Siegel Spaces
  4. Quotient Structure
  5. Riemannian Metric
  6. Riemannian Symmetric Spaces
  7. Proposed Approach
  8. Point-to-hyperplane Distances Based on the Quotient Structure of Siegel Spaces
  9. Hyperplanes
  10. Point-to-hyperplane Distance
  11. Product Spaces
  12. Point-to-hyperplane Distances Based on Vector-Valued Distances
  13. Hyperplanes
  14. Point-to-hyperplane Distance
  15. Neural Networks on Siegel Spaces
  16. ...and 10 more sections

Key Result

Proposition 3.3

The inner product $\langle \cdot,\cdot \rangle_{\mathbb{S}}$ agrees with the Riemannian distance, i.e., where $x,y \in X$, and the norm $\| \cdot \|_{\mathbb{S}}$ is induced by the inner product $\langle \cdot,\cdot \rangle_{\mathbb{S}}$. Furthermore, the inner product $\langle \cdot,\cdot \rangle_{\mathbb{S}}$ is invariant under the action of K, i.e., for any $k \in K$,

Figures (1)

  • Figure 1: The distance between a point $x \in \mathbb{R}^m$ and a Euclidean hyperplane $\mathcal{H}^E_{\xi,p}$.

Theorems & Definitions (13)

  • Definition 3.1: NguyenICLR25
  • Definition 3.2
  • Proposition 3.3
  • Definition 3.4
  • Theorem 3.5
  • Definition 3.6
  • Definition 3.7
  • Theorem 3.8
  • Definition 3.9
  • Definition 3.10
  • ...and 3 more