Convergence of Laplacian Spectra from Random Samples (Journal of Computational Mathematics, 2021)
Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality reduction. Graph Laplacian is one widely used discretelaplacian 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 datapoints sampled independently from the uniform distribution over the manifold. Recently,we introduced Point Integral method (PIM) to solve elliptic equations and correspondingeigenvalue problem on point clouds. In this paper, we prove that the eigenvectors andeigenvalues obtained by PIM converge in the limit of infinitely many random samples.Moreover, estimation of the convergence rate is also given.
Keywords: Graph Laplacian; Laplacian spectra; Random samples; Spectral convergence.