1-Lipschitz Neural Networks on Hadamard Manifolds

Forum topic · 小凯 · 2026-07-23

Summary

This arXiv paper (2507.17081) by Davide Murari, Marta Ghirardelli, and Ben Adcock constructs and analyzes a class of 1-Lipschitz neural networks on Hadamard manifolds. Controlling the Lipschitz constant is a standard technique for promoting robustness and stability in neural networks, but most existing strategies are designed for Euclidean spaces. The authors propose gradient-descent-type layers that are 1-Lipschitz and quasi-alpha-firmly nonexpansive, using Busemann functions and their gradient flows as core building blocks to design 1-Lipschitz geometry-preserving layers. They provide explicit constructions for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices. Two numerical experiments validate the approach: robust classification on the Poincare disk under hyperbolic perturbations, and masked Wishart covariance reconstruction, where an SPD-valued denoiser serves as a plug-and-play prior and outperforms static, purely data-driven, and Log-Euclidean denoising baselines.

Overview

Field: Machine Learning Authors: Davide Murari, Marta Ghirardelli, Ben Adcock Published: 2026-07-22 arXiv: 2507.17081

Abstract (English)

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- $\alpha$ -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 Poincare disk, and masked Wishart covariance reconstruction. On the Poincare disk, the proposed networks yield robust classifiers under hyperbolic perturbations. On the SPD manifold, we train SPD-valued denoisers and use them as plug-and-play priors for the masked Wishart covariance reconstruction problem. We show that the nonexpansive denoisers achieve improved results compared to static, purely data-driven, and Log-Euclidean denoising baselines, and we empirically test their convergence properties.

Key Contributions

A class of 1-Lipschitz, quasi- $\alpha$ -firmly nonexpansive gradient-descent-type layers on Hadamard manifolds
Busemann functions and their gradient flows as building blocks for geometry-preserving 1-Lipschitz layers
Explicit constructions for hyperbolic manifolds and SPD matrix manifolds
Experiments on robust classification (Poincare disk) and plug-and-play denoising for masked Wishart covariance reconstruction

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*Automatically collected on 2026-07-23*