1-Lipschitz Neural Networks on Hadamard Manifolds

2026-07-21

AI Summary

This work addresses the challenge of constructing strictly 1-Lipschitz neural networks on Hadamard manifolds—such as hyperbolic spaces and symmetric positive definite (SPD) manifolds—to ensure model robustness and stability. It extends the notion of 1-Lipschitz constraints to general Hadamard manifolds for the first time, introducing a novel non-expansive network layer based on Busemann functions and proving its quasi-α-strongly non-expansive property. By integrating gradient flows with geometry-preserving structures, the proposed method achieves robust classification against hyperbolic perturbations in the Poincaré disk and constructs an effective covariance denoiser on SPD manifolds. Empirical results demonstrate significant performance gains over static, data-driven, and Log-Euclidean baseline approaches.

Abstract

Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean spaces. In this work, we construct and analyze a class of 1-Lipschitz neural networks on Hadamard manifolds. Our layers are of gradient-descent type, $1$-Lipschitz, and quasi-$α$-firmly nonexpansive. The core building blocks of the proposed architecture are Busemann functions, and we exploit the properties of Busemann gradient flows to design $1$-Lipschitz geometry-preserving layers. We provide explicit constructions and examples for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices. We test the proposed architecture in two numerical experiments: robust classification on the Poincaré disk and masked-Wishart covariance reconstruction. On the Poincaré disk, the proposed networks yield robust classifiers under hyperbolic perturbations. On the SPD manifold, we train SPD-valued denoisers and adopt them as a Plug-and-Play prior for a masked-Wishart covariance reconstruction problem. We show improved results from the nonexpansive denoiser over static, data-only, and Log-Euclidean denoising baselines, and empirically test its convergence properties.

Problem

• 1-Lipschitz neural networks
• Hadamard manifolds
• robustness
• non-Euclidean geometry
• Busemann functions

Innovation

• 1-Lipschitz neural networks
• Hadamard manifolds
• Busemann functions
• nonexpansive layers
• geometric deep learning