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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| Published in | Applied and computational harmonic analysis Vol. 60; pp. 123 - 175 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.09.2022
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1063-5203 1096-603X |
| DOI | 10.1016/j.acha.2022.02.004 |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | García Trillos, Nicolás Calder, Jeff |
| Author_xml | – sequence: 1 givenname: Jeff surname: Calder fullname: Calder, Jeff email: jcalder@umn.edu organization: School of Mathematics, University of Minnesota, United States of America – sequence: 2 givenname: Nicolás orcidid: 0000-0002-7711-5901 surname: García Trillos fullname: García Trillos, Nicolás email: nicolasgarcia@stat.wisc.edu organization: Department of Statistics, University of Wisconsin-Madison, United States of America |
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| Cites_doi | 10.1073/pnas.0500334102 10.1007/978-1-4757-2201-7 10.1137/17M1134214 10.1137/18M1188999 10.1088/1361-6544/aae949 10.1016/j.acha.2006.03.004 10.1214/17-EJS1253 10.1016/j.acha.2016.09.003 10.1214/13-AOS1189 10.1137/18M1199241 10.1007/s00440-014-0583-7 10.1137/17M115222X 10.1007/s11222-007-9033-z 10.4171/jst/83 10.1016/j.tcs.2009.01.009 10.1214/009053607000000640 |
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| Keywords | Discrete to continuum Graph Laplacian Spectral convergence Rates of convergence Laplace-Beltrami operator |
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| Snippet | In this paper we improve the spectral convergence rates for graph-based approximations of weighted Laplace-Beltrami operators constructed from random data. We... |
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| SubjectTerms | Discrete to continuum Graph Laplacian Laplace-Beltrami operator Rates of convergence Spectral convergence |
| Title | Improved spectral convergence rates for graph Laplacians on ε-graphs and k-NN graphs |
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