Siegel Neural Networks

2025-11-12

AI Summary

This work addresses the unexplored problem of constructing discriminative neural networks on Siegel spaces—a prominent class of Riemannian symmetric spaces. We propose, for the first time, Riemannian-compatible multi-class logistic regression (MLR) and fully connected (FC) layers tailored to Siegel geometry. Methodologically, we leverage the quotient-space structure of Siegel manifolds and vector-valued distance metrics to embed classification decisions directly into their intrinsic geometry, thereby avoiding Euclidean approximations and ensuring strictly Riemannian-compliant forward propagation and parameter optimization. Our approach extends existing hyperbolic and SPD-manifold neural network frameworks, establishing the first geometric deep learning model grounded in Siegel space. Extensive experiments on radar clutter classification and graph node classification demonstrate state-of-the-art performance, validating the model’s effectiveness, generalization capability, and advantage in encoding geometric priors.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Security and privacy of machine learning and AI applications

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.

Problem

• Building neural networks on Siegel spaces for machine learning
• Developing multiclass logistic regression layers for Siegel spaces
• Applying Siegel neural networks to radar and node classification

Innovation

• Builds neural networks on Siegel spaces
• Uses quotient structure and vector-valued distance
• Applies to radar and node classification tasks