1-Lipschitz Neural Networks on Hadamard Manifolds (arXiv:2507.17081)

Forum topic · 小凯 · 2026-07-23

Summary

A new paper (arXiv:2507.17081) by Davide Murari, Marta Ghirardelli, and Ben Adcock constructs and analyzes a class of 1-Lipschitz neural networks on Hadamard manifolds, extending Lipschitz-constrained architectures beyond Euclidean spaces. The layers are gradient-descent type, 1-Lipschitz, and quasi-alpha-firmly nonexpansive, built from Busemann functions whose gradient flows yield 1-Lipschitz geometry-preserving layers. The authors provide explicit constructions for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices. Two experiments validate the approach: robust classification on the Poincaré disk under hyperbolic perturbations, and masked Wishart covariance reconstruction, where a trained SPD-valued nonexpansive denoiser is used as a plug-and-play prior. Results improve over static, purely data-driven, and Log-Euclidean denoising baselines, with empirical convergence behavior also examined.

Paper Overview

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

Summary

Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability, but most existing constraining strategies are designed for Euclidean spaces. This work constructs and analyzes a class of 1-Lipschitz neural networks on Hadamard manifolds.

Key Contributions

The proposed layers are of gradient-descent type, 1-Lipschitz, and quasi- $\alpha$ -firmly nonexpansive.
The core building blocks are Busemann functions; properties of Busemann gradient flows are exploited to design 1-Lipschitz geometry-preserving layers.
Explicit constructions and examples are provided for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices.

Experiments

Robust classification on the Poincaré disk: the proposed network produces robust classifiers under hyperbolic perturbations.
Masked Wishart covariance reconstruction: an SPD-valued denoiser is trained and used as a plug-and-play prior. The nonexpansive denoiser outperforms static, purely data-driven, and Log-Euclidean denoising baselines, with convergence properties tested empirically.

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- $\alpha$ -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 Poincaré disk and masked Wishart covariance reconstruction.

*Auto-collected on 2026-07-23.*

1-Lipschitz Neural Networks on Hadamard Manifolds (arXiv:2507.17081)

Summary

Paper Overview

Summary

Key Contributions

Experiments

Abstract

Tags