Lipschitz regularity of graph Laplacians on random data clouds

TL;DR

This paper establishes Lipschitz regularity for solutions of graph-based elliptic PDEs on random data clouds, providing theoretical guarantees for the smoothness of graph Laplacian eigenvectors in data analysis.

Contribution

It introduces probabilistic Lipschitz regularity estimates for graph PDE solutions and eigenvectors, connecting discrete graph properties to continuum manifold regularity.

Findings

High probability Lipschitz estimates for graph Poisson solutions

Lipschitz regularity of graph Laplacian eigenvectors with explicit constants

Convergence rates for eigenvectors towards continuum eigenfunctions

Abstract

In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in data analysis in the context of graph-based learning and contains, as important examples, the equations satisfied by graph Laplacian eigenvectors. In particular, we prove high probability interior and global Lipschitz estimates for solutions of graph Poisson equations. Our results can be used to show that graph Laplacian eigenvectors are, with high probability, essentially Lipschitz regular with constants depending explicitly on their corresponding eigenvalues. Our analysis relies on a probabilistic coupling argument of suitable random walks at the continuum level, and an interpolation method for extending functions on random point clouds to the continuum manifold. As a byproduct of our general regularity results, we obtain high probability $L^\infty$ and approximate $\mathcal{C}^{0,1}$ convergence rates for the convergence of graph Laplacian eigenvectors towards eigenfunctions of the corresponding weighted Laplace-Beltrami operators. The convergence rates we obtain scale like the $L^2$-convergence rates established by two of the authors in previous work.