A kernel-based analysis of Laplacian Eigenmaps

2024/02/26 by Martin Wahl, Wahl, Martin
Biochemistry, Genetics and Molecular Biology · #15A42 #35K08 #47A55 #47D07 #60B20 #62H25 #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Probability (math.PR) #RNA Research and Splicing #Spectral Theory (math.SP) #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2402.16481

openalex publication_date 2024/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given i.i.d. observations uniformly distributed on a closed manifold M⊆ ℝ^p, we study the spectral properties of the associated empirical graph Laplacian based on a Gaussian kernel. Our main results are non-asymptotic error bounds, showing that the eigenvalues and eigenspaces of the empirical graph Laplacian are close to the eigenvalues and eigenspaces of the Laplace-Beltrami operator of M. In our analysis, we connect the empirical graph Laplacian to kernel principal component analysis, and consider the heat kernel of M as reproducing kernel feature map. This leads to novel points of view and allows to leverage results for empirical covariance operators in infinite dimensions.

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