Spectral Convergence of Symmetrized Graph Laplacian on manifolds with boundary

2021/10/13 by J. Wilson Peoples, John Harlim, Peoples, J. Wilson +1 · 2 citations
Computer Science · Mathematics · #05C50 #34L15 #34L16 #47N40 #58J50 #60D05 #65N25 #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2110.06988

openalex publication_date 2021/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the spectral convergence of a symmetrized Graph Laplacian matrix induced by a Gaussian kernel evaluated on pairs of embedded data, sampled from a manifold with boundary, a sub-manifold of ℝ^m. Specifically, we deduce the convergence rates for eigenpairs of the discrete Graph-Laplacian matrix to the eigensolutions of the Laplace-Beltrami operator that are well-defined on manifolds with boundary, including the homogeneous Neumann and Dirichlet boundary conditions. For the Dirichlet problem, we deduce the convergence of the truncated Graph Laplacian, which is recently numerically observed in applications, and provide a detailed numerical investigation on simple manifolds. Our method of proof relies on the min-max argument over a compact and symmetric integral operator, leveraging the RKHS theory for spectral convergence of integral operator and a recent pointwise asymptotic result of a Gaussian kernel integral operator on manifolds with boundary.

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