1-Lipschitz Neural Networks on Hadamard Manifolds
2026-07-21 • Machine Learning
Machine Learning
AI summaryⓘ
The authors develop a new type of neural network that maintains a strict control over how much it can stretch or distort data, but specifically designed for curved spaces called Hadamard manifolds. They build these networks using special mathematical functions called Busemann functions, ensuring the layers don't expand distances more than necessary. Their approach is applied to tasks like classifying data on a hyperbolic space and improving covariance matrix denoising, showing better performance than traditional methods. This work helps extend robust neural network design to more complex geometric settings.
Lipschitz constantHadamard manifoldBusemann functionGradient flow1-Lipschitz networkHyperbolic spaceSPD matrix manifoldCovariance matrix reconstructionPlug-and-Play prior
Authors
Davide Murari, Marta Ghirardelli, Ben Adcock, Elena Celledoni, Brynjulf Owren, Carola-Bibiane Schönlieb
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.