Convergence of Laplacian Spectra from Point Clouds

Abstract: The spectral structure of the Laplacian-Beltrami operator (LBO) on manifolds has been widely used in many applications, include spectral clustering, dimensionality reduction, mesh smoothing, compression and editing, shape segmentation, matching and parameterization, and so on. Typically, the underlying Riemannian manifold is unknown and often given by a set of sample points. The spectral structure of the LBO is estimated from some discrete Laplace operator constructed from this set of sample points. In our previous papers, we proposed the point integral method to discretize the LBO from point clouds, which is also capable to solve the eigenproblem. Then one fundmental issue is the convergence of the eigensystem of the discrete Laplacian to that of the LBO. In this paper, for compact manifolds isometrically embedded in Euclidean spaces possibly with boundary, we show that the eigenvalues and the eigenvectors obtained by the point integral method converges to the eigenvalues and the eigenfunctions of the LBO with the Neumann boundary, and in addition, we give an estimate of the convergence rate. This result provides a solid mathematical foundation for the point integral method in the computation of Laplacian spectra from point clouds.