  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.
