Spectral convergence of graph Laplacian and heat kernel reconstruction in L from random samples

David B. Dunson, Hau Tieng Wu, Nan Wu

Research output: Contribution to journalArticlepeer-review

Abstract

In the manifold setting, we provide a series of spectral convergence results quantifying how the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator in the L sense. Based on these results, convergence of the proposed heat kernel approximation algorithm, as well as the convergence rate, to the exact heat kernel is guaranteed. To our knowledge, this is the first work exploring the spectral convergence in the L sense and providing a numerical heat kernel reconstruction from the point cloud with theoretical guarantees.

Original languageEnglish (US)
Pages (from-to)282-336
Number of pages55
JournalApplied and Computational Harmonic Analysis
Volume55
DOIs
StatePublished - Nov 2021

Keywords

  • Graph Laplacian
  • Heat kernel
  • Laplace-Beltrami operator
  • Manifold learning

ASJC Scopus subject areas

  • Applied Mathematics

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