Siegel Neural Networks

TL;DR

This paper introduces a novel neural network framework on Siegel spaces, a type of Riemannian symmetric space, enabling effective classification in applications like radar clutter and node classification.

Contribution

It develops the first discriminative neural network layers for Siegel spaces, leveraging their quotient structure and vector-valued distances, advancing representation learning on these spaces.

Findings

Achieved state-of-the-art results on radar clutter classification

Demonstrated superior performance on node classification tasks

Validated the effectiveness of Siegel neural networks across datasets

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.