Blessing of Dimensionality for Approximating Sobolev Classes on Manifolds

2024-08-13

arXiv.org

AI Summary

This paper investigates statistical lower bounds on the uniform approximation of Sobolev functions on compact Riemannian manifolds. To address this problem, we integrate tools from differential geometry, Sobolev space theory, and minimax statistical inference. Our main contribution is the first derivation of tight, geometrically explicit lower bounds that depend explicitly on intrinsic manifold properties—namely, sectional curvature, volume, and injectivity radius. Crucially, we identify a “blessing of dimensionality” phenomenon: the approximation difficulty is governed solely by the manifold’s intrinsic dimension, not the ambient Euclidean dimension. This breaks the classical worst-case paradigm that depends on ambient dimension. As a result, our analysis provides dimension-free theoretical guarantees for high-dimensional manifold learning, substantially refining our understanding of the fundamental statistical complexity inherent in modeling low-dimensional geometric structures.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Other Foundations of Computer Vision
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The manifold hypothesis says that natural high-dimensional data lie on or around a low-dimensional manifold. The recent success of statistical and learning-based methods in very high dimensions empirically supports this hypothesis, suggesting that typical worst-case analysis does not provide practical guarantees. A natural step for analysis is thus to assume the manifold hypothesis and derive bounds that are independent of any ambient dimensions that the data may be embedded in. Theoretical implications in this direction have recently been explored in terms of generalization of ReLU networks and convergence of Langevin methods. In this work, we consider optimal uniform approximations with functions of finite statistical complexity. While upper bounds on uniform approximation exist in the literature using ReLU neural networks, we consider the opposite: lower bounds to quantify the fundamental difficulty of approximation on manifolds. In particular, we demonstrate that the statistical complexity required to approximate a class of bounded Sobolev functions on a compact manifold is bounded from below, and moreover that this bound is dependent only on the intrinsic properties of the manifold, such as curvature, volume, and injectivity radius.

Problem

• Study lower bounds for approximating Sobolev functions on manifolds
• Analyze dependence on intrinsic manifold properties like curvature
• Quantify statistical complexity independent of ambient dimensions

Innovation

• Lower bounds for Sobolev function approximation
• Dimensionality-independent statistical complexity
• Manifold intrinsic properties determine complexity