Convergence of Laplacian spectra from random samples
This provides a theoretical foundation for manifold learning methods using graph Laplacians under non-uniform sampling, extending previous uniform-sampling results.
The paper proves that eigenvectors and eigenvalues from the Point Integral Method (PIM) converge to those of the Laplace-Beltrami operator on a manifold, even for non-uniform sampling distributions, and provides a convergence rate estimate.
Eigenvectors and eigenvalues of discrete graph Laplacians are often used for manifold learning and nonlinear dimensionality reduction. It was previously proved by Belkin and Niyogi 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 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. We have established a unified framework to approximate the elliptic differential operators 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 independently from a distribution (not necessarily to be uniform distribution). Moreover, one estimate of the rate of the convergence is also given.