aiqbench analysis
This research extends neural network robustness constraints from Euclidean spaces to Hadamard manifolds, specifically targeting hyperbolic geometry and symmetric positive definite matrices. By leveraging Busemann functions, the authors demonstrate improved performance in tasks like masked-Wishart covariance reconstruction under non-Euclidean perturbations. The work advances theoretical machine learning by providing explicit constructions for 1-Lipschitz layers that preserve complex geometric structures essential for specific data distributions.
1-Lipschitz Neural Networks on Hadamard Manifolds
arXiv:2607.19335v1 Announce Type: cross 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 Poincar\'e disk and masked-Wishart covariance reconstruction. On the Poincar\'e 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.
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. 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.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2607.19335 [math.NA]
(or arXiv:2607.19335v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2607.19335 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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