Neural Networks on Symmetric Spaces of Noncompact Type

2026-01-06 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Software Development & Engineering · Depth: Expert, extended

Summary

This paper introduces a novel approach for developing neural networks on symmetric spaces of noncompact type, which include hyperbolic spaces and symmetric positive definite (SPD) manifolds. The core of the method is a unified formulation of the distance from a point to a hyperplane within these spaces. The authors derive a closed-form expression for this point-to-hyperplane distance, particularly for higher-rank symmetric spaces equipped with G-invariant Riemannian metrics. This derived distance is then utilized to design fundamental neural network building blocks, specifically fully-connected (FC) layers and an attention mechanism. The proposed approach is validated across challenging benchmarks, demonstrating its efficacy in tasks such as image classification, electroencephalogram (EEG) signal classification, image generation, and natural language inference, often outperforming existing hyperbolic neural network models.

Key takeaway

For research scientists working on machine learning in non-Euclidean geometries, this work provides a robust framework for building neural networks on symmetric spaces of noncompact type. You should consider integrating the proposed unified point-to-hyperplane distance into your models, especially for tasks involving hierarchical data or matrix manifolds. The demonstrated performance improvements in image, EEG, and NLP tasks suggest that these new FC layers and attention mechanisms can enhance model accuracy and stability, particularly when dealing with datasets exhibiting strong hierarchical structures.

Key insights

A unified point-to-hyperplane distance enables novel neural network architectures on noncompact symmetric spaces.

Principles

• Busemann functions generalize Euclidean coordinates for non-Euclidean spaces.
• Horocycles serve as symmetric space analogs of Euclidean hyperplanes.
• G-invariant metrics offer better scalability for high-dimensional inputs.

Method

The method defines hyperplanes using Busemann functions and derives a point-to-hyperplane distance. This distance then informs the design of FC layers and an attention mechanism for neural networks on symmetric spaces.

In practice

• Apply the unified distance to improve existing hyperbolic neural networks.
• Utilize G-invariant metrics for FC layers with high-dimensional SPD matrices.
• Implement the proposed attention mechanism for sequence processing on manifolds.

Topics

• Symmetric Space Neural Networks
• Point-to-Hyperplane Distance
• Hyperbolic Neural Networks
• SPD Manifolds
• G-invariant Metrics

Code references

• nguyenxuanson10/symspaces-ic
• sohata24/nli
• dalab/hyperbolic_nn
• nguyenxuanson10/symspaces-nc

Best for: Research Scientist, AI Researcher, AI Scientist, Deep Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.