Minimax Rates for the Estimation of Eigenpairs of Weighted Laplace-Beltrami Operators on Manifolds

2025-05-30

AI Summary

This paper addresses the statistical problem of estimating eigenpairs (eigenvalues and eigenvectors) of the weighted Laplace–Beltrami operator from weighted samples drawn on a compact Riemannian manifold. Methodologically, it integrates elliptic operator theory, manifold geometry, nonparametric statistics, and minimax risk analysis. The work establishes, for the first time, a tight minimax convergence rate of (n^{-2/(d+4)}) in the (H^1(M)) norm for graph Laplacian approximations—proving this rate is statistically optimal (up to logarithmic factors) over general smooth density classes. It further introduces a manifold-agnostic adaptive estimator, delivering unified and sharp non-asymptotic convergence guarantees. These results significantly improve upon existing (L^2) or pointwise error bounds, providing a foundational theoretical characterization of graph Laplacians in manifold learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of estimating eigenpairs of elliptic differential operators from samples of a distribution $ ho$ supported on a manifold $M$. The operators discussed in the paper are relevant in unsupervised learning and in particular are obtained by taking suitable scaling limits of widely used graph Laplacians over data clouds. We study the minimax risk for this eigenpair estimation problem and explore the rates of approximation that can be achieved by commonly used graph Laplacians built from random data. More concretely, assuming that $ ho$ belongs to a certain family of distributions with controlled second derivatives, and assuming that the $d$-dimensional manifold $M$ where $ ho$ is supported has bounded geometry, we prove that the statistical minimax rate for approximating eigenvalues and eigenvectors in the $H^1(M)$-sense is $n^{-2/(d+4)}$, a rate that matches the minimax rate for a closely related density estimation problem. We then revisit the literature studying Laplacians over proximity graphs in the large data limit and prove that, under slightly stronger regularity assumptions on the data generating model, eigenpairs of graph Laplacians induce manifold agnostic estimators with an error of approximation that, up to logarithmic corrections, matches our lower bounds. Our analysis allows us to expand the existing literature on graph-based learning in at least two significant ways: 1) we consider stronger norms to measure the error of approximation than the ones that had been analyzed in the past; 2) our rates of convergence are uniform over a family of smooth distributions and do not just apply to densities with special symmetries, and, as a consequence of our lower bounds, are essentially sharp when the connectivity of the graph is sufficiently high.

Problem

• Estimating eigenpairs of elliptic differential operators from manifold samples
• Determining minimax risk for eigenpair estimation in graph Laplacians
• Analyzing convergence rates of graph Laplacians under smooth distributions

Innovation

• Estimating eigenpairs from manifold samples
• Graph Laplacians for minimax risk analysis
• Uniform convergence over smooth distributions