Convergence of Laplacian Spectra from Random Samples

Zuoqiang Shi Tsinghua University

Numerical Analysis and Scientific Computing mathscidoc: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
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@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.
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