Graph approximations to the Laplacian spectra

TL;DR

This paper demonstrates that the spectra of Laplace-Beltrami operators on manifolds and metric-measure spaces can be efficiently approximated using graph Laplacians on proximity graphs, enabling faster spectral computations.

Contribution

It introduces a method to approximate the Laplace-Beltrami spectrum on manifolds and metric spaces via graph Laplacians, extending previous approaches to more general spaces.

Findings

Spectra of Laplace-Beltrami operators can be approximated by graph Laplacians.

The approximation is fast and applicable to various spaces.

Method works for manifolds with boundary and glued metric spaces.

Abstract

I prove that the spectrum of the Laplace-Beltrami operator with the Neumann boundary condition on a compact Riemannian manifold with boundary admits a fast approximation by the spectra of suitable graph Laplacians on proximity graphs on the manifold, and similar graph approximation works for metric-measure spaces glued out of compact Riemannian manifolds of the same dimension.