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October 20, 2025Open Access

Learning Beyond Euclid: Curvature-Adaptive Generalization for Neural Networks on Manifolds

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New generalization bounds for neural networks incorporate manifold-specific properties like curvature and volume growth.
Our geometric approach leads to improved Rademacher complexity bounds for Lipschitz neural networks defined on compact manifolds.
The results demonstrate that existing Euclidean bounds are recovered when curvature is zero, showing how intrinsic geometry can shape learning.
In negatively curved spaces, the complexity is higher due to exponential volume growth, while positive curvature acts as a regularizing factor.

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This analysis demonstrates generalization improvements in neural networks on Riemannian manifolds, suggesting curvature influences learning capacity.

Kamal Sarkar (2025) studied this question.

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This analysis demonstrates generalization improvements in neural networks on Riemannian manifolds, suggesting curvature influences learning capacity.