Spectral Convergence of the connection Laplacian from random samples

TL;DR

This paper proves that the eigenvectors and eigenvalues of connection Laplacians estimated from random samples converge to their continuous counterparts on the manifold, extending previous spectral convergence results to more general settings.

Contribution

It introduces a unified framework for approximating various connection Laplacians on manifolds and proves their spectral convergence from random samples, including non-uniform distributions and manifolds with boundary.

Findings

Eigenvectors and eigenvalues of connection Laplacians converge with infinite samples

Framework applies to non-uniform sampling and manifolds with boundary

Extends spectral convergence results to broader classes of Laplacians

Abstract

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Vector Diffusion Maps and showed that the connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In this paper, we present a unified framework for approximating other connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many independent random samples. We generalize the spectral convergence results to the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.