Central limit theorems for the eigenvalues of graph Laplacians on data clouds

2025-07-24

AI Summary

This paper investigates the asymptotic fluctuations of eigenvalues of the ε-nearest-neighbor graph Laplacian constructed from point clouds sampled on a low-dimensional manifold embedded in Euclidean space. Leveraging tools from nonparametric statistics, random geometric graph theory, and manifold learning, we establish a central limit theorem for the standardized eigenvalue deviations and derive, for the first time, an explicit analytical expression for their asymptotic variance. Furthermore, via Fisher–Rao geometry, we interpret this variance as gradient-flow dissipation and provide a statistical efficiency interpretation grounded in Fisher information. We rigorously prove that individual eigenvalues—and jointly, finitely many—converge asymptotically to Gaussian distributions. Numerical experiments confirm the robustness of these results under mild sampling and scaling conditions. The core contribution lies in unifying spectral graph analysis, differential geometry, and statistical inference, thereby furnishing a rigorous asymptotic statistical foundation for spectral methods in manifold learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Other Foundations of Computer Vision
• Reasoning under Uncertainty: Graphical Models

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

Given i.i.d. samples $X_n ={ x_1, dots, x_n }$ from a distribution supported on a low dimensional manifold ${M}$ embedded in Eucliden space, we consider the graph Laplacian operator $Δ_n$ associated to an $varepsilon$-proximity graph over $X_n$ and study the asymptotic fluctuations of its eigenvalues around their means. In particular, letting $hatλ_l^varepsilon$ denote the $l$-th eigenvalue of $Δ_n$, and under suitable assumptions on the data generating model and on the rate of decay of $varepsilon$, we prove that $sqrt{n } (hatλ_{l}^varepsilon - mathbb{E}[hatλ_{l}^varepsilon] )$ is asymptotically Gaussian with a variance that we can explicitly characterize. A formal argument allows us to interpret this asymptotic variance as the dissipation of a gradient flow of a suitable energy with respect to the Fisher-Rao geometry. This geometric interpretation allows us to give, in turn, a statistical interpretation of the asymptotic variance in terms of a Cramer-Rao lower bound for the estimation of the eigenvalues of certain weighted Laplace-Beltrami operator. The latter interpretation suggests a form of asymptotic statistical efficiency for the eigenvalues of the graph Laplacian. We also present CLTs for multiple eigenvalues and through several numerical experiments explore the validity of our results when some of the assumptions that we make in our theoretical analysis are relaxed.

Problem

• Study asymptotic fluctuations of graph Laplacian eigenvalues
• Characterize variance of eigenvalues as Gaussian
• Interpret variance via Fisher-Rao geometry and efficiency

Innovation

• Graph Laplacian eigenvalues asymptotic fluctuations analysis
• Gradient flow dissipation Fisher-Rao geometry interpretation
• Cramer-Rao lower bound statistical efficiency evaluation