Numerical Analysis and Scientific Computingmathscidoc:1709.25002
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 Niyogi [4] that 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 in nitely many data points sampled independently from the uniform distribution over the
manifold. Recently, we introduced Point Integral method (PIM) [15] 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 in nitely many random samples. Moreover, one
estimate of the rate of the convergence is given.
Graph Laplacian; Laplacian spectra; random samples; convergence rate
@inproceedings{zuoqiangconvergence,
title={Convergence of Laplacian Spectra from Random Samples},
author={Zuoqiang Shi},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170925160933933889818},
}
Zuoqiang Shi. Convergence of Laplacian Spectra from Random Samples. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170925160933933889818.