Spectral convergence of graph Laplacians with Ricci curvature bounds and in non-collapsed Ricci limit spaces

TL;DR

This paper provides probabilistic bounds on the eigenvalues and eigenfunctions of graph Laplacians constructed from data on manifolds with Ricci curvature bounds, extending results to non-smooth Ricci limit spaces.

Contribution

It extends spectral convergence results of graph Laplacians to non-collapsed Ricci limit spaces with curvature and volume bounds, including non-smooth points.

Findings

Quantitative high-probability bounds on eigenvalues and eigenfunctions

Extension of spectral approximation to Ricci limit spaces

Applicability to non-smooth geometric settings

Abstract

This paper establishes quantitative high-probability bounds on the eigenvalues and eigenfunctions of $\epsilon$-neighborhood graph Laplacians constructed from i.i.d. random variables on $m$-dimensional closed Riemannian manifolds $(M,g)$ that satisfy a uniform lower Ricci curvature bound $\operatorname{Ric}_g\ge -(m-1)K$, a positive lower volume bound, and an upper diameter bound. These results extend to non-collapsed Ricci limit spaces that are measured Gromov-Hausdorff limits of such manifolds, and the bounds give a spectral approximation of weighted Laplacians on manifolds with non-smooth points.