2025/07/24 by LI Cheng-hui, Nicolás García Trillos, Li, Chenghui +5 · 1 citation
Mathematics · Physics and Astronomy · #62G20 60F05 58J50 35P15 68R10 60D05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Geometric Analysis and Curvature Flows #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Probability (math.PR) #Random Matrices and Applications #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2507.18803
openalex publication_date 2025/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given i.i.d. samples Xn =\ x1, …, xn \ from a distribution supported on a low dimensional manifold M embedded in Eucliden space, we consider the graph Laplacian operator Δn associated to an ε-proximity graph over Xn and study the asymptotic fluctuations of its eigenvalues around their means. In particular, letting λlε denote the l-th eigenvalue of Δn, and under suitable assumptions on the data generating model and on the rate of decay of ε, we prove that √(n ) (λlε - 𝔼[λlε] ) 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.