1-Lipschitz Neural Networks on Hadamard Manifolds

• URL: http://arxiv.org/abs/2607.19335v1
• Date: Tue, 21 Jul 2026 17:54:34 GMT
• Title: 1-Lipschitz Neural Networks on Hadamard Manifolds
• Abstract summary: Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability.<n>We construct and analyze a class of 1-Lipschitz neural networks on Hadamard manifold.<n>We show improved results from the nonexpansive denoiser over static, data-only, and Log-Euclidean denoising baselines.
• Score: 19.375426747762518
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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