A graph discretization of the Laplace-Beltrami operator

TL;DR

This paper demonstrates that the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold can be effectively approximated using a graph Laplacian constructed from a proximity graph on an epsilon-net, bridging continuous and discrete spectral analysis.

Contribution

The paper introduces a graph discretization method for the Laplace-Beltrami operator, providing a new approach to approximate its spectral properties using graph Laplacians.

Findings

Eigenvalues of the graph Laplacian approximate those of the Laplace-Beltrami operator.

Eigenfunctions of the graph Laplacian converge to the eigenfunctions of the Laplace-Beltrami operator.

The method applies to Riemannian manifolds using a suitably weighted proximity graph.

Abstract

We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.