Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs

TL;DR

This paper improves the theoretical understanding of how quickly graph Laplacians built from random data converge to the true Laplace-Beltrami operator, providing optimal rates for various graph constructions.

Contribution

It establishes that spectral convergence rates match pointwise consistency rates for a broad class of graph Laplacians, including epsilon-graphs and k-NN graphs.

Findings

Eigenvalues and eigenvectors converge at rate O(n^{-1/(m+4)})

Results apply to epsilon-graphs and k-NN graphs

Convergence rates match pointwise consistency rates

Abstract

In this paper we improve the spectral convergence rates for graph-based approximations of Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity $\varepsilon$, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of the Laplace-Beltrami operator at a rate of $O(n^{-1/(m+4)})$, up to log factors, where $m$ is the manifold dimension and $n$ is the number of vertices in the graph. Our approach is general and allows us to analyze a large variety of graph constructions that include $\varepsilon$-graphs and $k$-NN graphs.