哈达玛流形上的1-利普希茨神经网络

中文摘要 AI 辅助

控制神经网络的利普希茨常数是提升鲁棒性和稳定性的标准方法。现有多数约束策略是针对欧几里得空间设计的。本文在哈达玛流形上构建并分析了一类1-利普希茨神经网络。其层为梯度下降型、1-利普希茨且拟α-强非扩张的。核心构建块是布泽曼函数，利用其梯度流属性设计1-利普希茨保形层。给出双曲流形和对称正定矩阵流形的显式构造与示例。通过两个数值实验测试该架构：庞加莱圆盘上的鲁棒分类和掩码威沙特协方差重构。结果表明该网络在双曲扰动下产生鲁棒分类器，在SPD流形上训练的去噪器有更好效果并测试了收敛性。

英文摘要

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-$α$-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.

发表机构

• Department of Applied Mathematics and Theoretical Physics, University of Cambridge（应用数学与理论物理系，剑桥大学）
• Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU)（数学科学系，挪威科学技术大学（NTNU））
• Department of Mathematics, Simon Fraser University（数学系，西蒙·弗雷泽大学）

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