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

• URL: http://arxiv.org/abs/2507.02999v1
• Date: Tue, 01 Jul 2025 23:16:49 GMT
• Title: Learning Beyond Euclid: Curvature-Adaptive Generalization for Neural Networks on Manifolds
• Authors: Krisanu Sarkar,
• Abstract summary: Existing generalization theories often rely on complexity measures derived from Euclidean geometry, which fail to account for intrinsic structure of non-Euclidean spaces.<n>We derive number bounds that explicitly incorporate manifold-specific properties such as sectional curvature, volume growth, and injectivity radius.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Adaptive Riemannian Graph Neural Networks [29.859977834688625]
  We introduce a novel framework that learns a continuous and anisotropic metric tensor field over the graph.<n>It allows each node to determine its optimal local geometry, enabling the model to fluidly adapt to the graph's structural landscape.<n>Our method demonstrates superior performance on both homophilic and heterophilic benchmark geometries.
  arXiv  Detail & Related papers  (2025-08-04T16:55:02Z)
• Understanding In-Context Learning on Structured Manifolds: Bridging Attention to Kernel Methods [48.038668788625465]
  In-context learning (ICL) has achieved remarkable success in natural language and vision domains.<n>In this work, we initiate a theoretical study of ICL for regression of H"older functions on manifold.<n>Our findings provide foundational insights into the role of geometry in ICL and novels tools to study ICL of nonlinear models.
  arXiv  Detail & Related papers  (2025-06-12T17:56:26Z)
• Decoder ensembling for learned latent geometries [15.484595752241122]
  We show how to easily compute geodesics on the associated expected manifold. We find this simple and reliable, thereby coming one step closer to easy-to-use latent geometries.
  arXiv  Detail & Related papers  (2024-08-14T12:35:41Z)
• Riemannian Residual Neural Networks [58.925132597945634]
  We show how to extend the residual neural network (ResNet) ResNets have become ubiquitous in machine learning due to their beneficial learning properties, excellent empirical results, and easy-to-incorporate nature when building varied neural networks.
  arXiv  Detail & Related papers  (2023-10-16T02:12:32Z)
• Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds [77.4346324549323]
  We show that a step size agnostic to the curvature of the manifold achieves a curvature-independent and linear last-iterate convergence rate. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence has not been considered before.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Exploring Data Geometry for Continual Learning [64.4358878435983]
  We study continual learning from a novel perspective by exploring data geometry for the non-stationary stream of data. Our method dynamically expands the geometry of the underlying space to match growing geometric structures induced by new data. Experiments show that our method achieves better performance than baseline methods designed in Euclidean space.
  arXiv  Detail & Related papers  (2023-04-08T06:35:25Z)
• kHGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning [39.25873010585029]
  This study endeavors to explore the curvature between discrete structure and continuous learning space, aiming at encoding the message conveyed by the network topology in the learning process. A curvature-aware hyperbolic graph convolutional neural network, kappaHGCN, is proposed, which utilizes the curvature to guide message passing and improve long-range propagation.
  arXiv  Detail & Related papers  (2022-12-04T10:45:42Z)
• A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations [77.86290991564829]
  Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. We study a particular sequence of maps between manifold, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between implementing neural networks of practical interest.
  arXiv  Detail & Related papers  (2021-12-17T11:43:30Z)
• Deep Networks Provably Classify Data on Curves [12.309532551321334]
  We study a model problem that uses a deep fully-connected neural network to classify data drawn from two disjoint smooth curves on the unit sphere. We prove that when (i) the network depth is large to certain properties that set the difficulty of the problem and (ii) the network width and number of samples is intrinsic in the relative depth, randomly-d gradient descent quickly learns to correctly classify all points on the two curves with high probability.
  arXiv  Detail & Related papers  (2021-07-29T20:40:04Z)
• Semi-Riemannian Graph Convolutional Networks [36.09315878397234]
  We develop a principled Semi-Riemannian GCN that first models data in semi-Riemannian manifold of constant nonzero curvature. Our method provides a geometric inductive bias that is sufficiently flexible to model mixed heterogeneous topologies like hierarchical graphs with cycles.
  arXiv  Detail & Related papers  (2021-06-06T14:23:34Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.