Improved spectral convergence rates for graph Laplacians on ε-graphs and k-NN graphs

In this paper we improve the spectral convergence rates for graph-based approximations of weighted Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the...

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Veröffentlicht in:Applied and computational harmonic analysis Jg. 60; S. 123 - 175
Hauptverfasser: Calder, Jeff, García Trillos, Nicolás
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.09.2022
Schlagworte:
Discrete to continuum
Graph Laplacian
Laplace-Beltrami operator
Rates of convergence
Spectral convergence
Discrete to continuum
Graph Laplacian
Spectral convergence
Rates of convergence
Laplace-Beltrami operator
ISSN:1063-5203, 1096-603X
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Zusammenfassung:In this paper we improve the spectral convergence rates for graph-based approximations of weighted Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity ε, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of a weighted Laplace-Beltrami operator at a rate of O(n−1/(m+4)), up to log factors, where m is the manifold dimension and n is the number of vertices in the graph. Our approach is general and allows us to analyze a large variety of graph constructions that include ε-graphs and k-NN graphs. We also present the results of numerical experiments analyzing convergence rates on the two dimensional sphere.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2022.02.004