Year: 2020
Author: Wenqi Tao, Zuoqiang Shi
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 6 : pp. 952–984
Abstract
Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality reduction. Graph Laplacian is one widely used discrete laplacian on point cloud. It was previously proved by Belkin and Niyogithat the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Point Integral method (PIM) to solve elliptic equations and corresponding eigenvalue problem on point clouds. In this paper, we prove that the eigenvectors and eigenvalues obtained by PIM converge in the limit of infinitely many random samples. Moreover, estimation of the convergence rate is also given.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2008-m2018-0232
Journal of Computational Mathematics, Vol. 38 (2020), Iss. 6 : pp. 952–984
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 33
Keywords: Graph Laplacian Laplacian spectra Random samples Spectral convergence.