[论文] 1-Lipschitz Neural Networks on Hadamard Manifolds

论文概要

研究领域: ML 作者: Davide Murari, Marta Ghirardelli, Ben Adcock 发布时间: 2026-07-22 arXiv: 2507.17081

中文摘要

控制神经网络的Lipschitz常数是提升鲁棒性和稳定性的标准方法。大多数现有约束策略为欧几里得空间设计。本工作中，我们在Hadamard流形上构造并分析了一类1-Lipschitz神经网络。我们的层具有梯度下降类型、1-Lipschitz和准$\alpha$-牢固非扩张性。所提出架构的核心构建块是Busemann函数，我们利用Busemann梯度流的性质设计1-Lipschitz保几何层。我们为双曲流形和对称正定（SPD）矩阵流形提供了显式构造和示例。我们在两个数值实验中测试所提出的架构：Poincaré圆盘上的鲁棒分类和掩码Wishart协方差重建。在Poincaré圆盘上，所提出的网络在双曲扰动下产生鲁棒分类器。在SPD流形上，我们训练SPD值去噪器，并将其用作掩码Wishart协方差重建问题的即插即用先验。我们展示了非扩张去噪器相比静态、纯数据和Log-Euclidean去噪基线的改进结果，并经验性测试其收敛性质。

原文摘要

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 Poin...

--- *自动采集于 2026-07-23*

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