为流形上神经网络提出基于曲率的泛化理论，更精准预测学习性能。
Learning Beyond Euclid: Curvature-Adaptive Generalization for Neural Networks on Manifolds
- 引入流形曲率、体积增长等几何特性修正覆盖数
- 在负曲率空间中揭示指数级复杂度提升，在正曲率中体现正则化作用
- 适用于高维嵌入低维流形的数据，如形状、图像等结构化数据
本文为定义在黎曼流形上的神经网络建立了新的泛化界。现有理论多依赖欧氏几何的复杂度度量，无法刻画非欧空间的内在结构。我们的分析引入几何修正：推导出显式包含截面曲率、体积增长和可浸入半径等流形特性的覆盖数界。这些几何调整使得在紧致流形上定义的Lipschitz神经网络的Rademacher复杂度界更加紧致。当曲率为零时，结果退化为标准欧氏情形；而在数据位于高维环境中的低维弯曲流形上，泛化保证显著改进。我们在负曲率空间中验证了界的有效性，其指数级体积增长导致复杂度明确上升；在正曲率空间中，曲率起到正则化作用。该框架从理论上揭示了内在几何如何影响学习能力，为结构化数据上的深度学习提供原理支撑与实践指导。
原文摘要 · Abstract (English)
In this work, we develop new generalization bounds for neural networks trained on data supported on Riemannian manifolds. Existing generalization theories often rely on complexity measures derived from Euclidean geometry, which fail to account for the intrinsic structure of non-Euclidean spaces. Our analysis introduces a geometric refinement: we derive covering number bounds that explicitly incorporate manifold-specific properties such as sectional curvature, volume growth, and injectivity radius. These geometric corrections lead to sharper Rademacher complexity bounds for classes of Lipschitz neural networks defined on compact manifolds. The resulting generalization guarantees recover standard Euclidean results when curvature is zero but improve substantially in settings where the data lies on curved, low-dimensional manifolds embedded in high-dimensional ambient spaces. We illustrate the tightness of our bounds in negatively curved spaces, where the exponential volume growth leads to provably higher complexity, and in positively curved spaces, where the curvature acts as a regularizing factor. This framework provides a principled understanding of how intrinsic geometry affects learning capacity, offering both theoretical insight and practical implications for deep learning on structured data domains.
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