On empirical Hodge Laplacians under the manifold hypothesis
Jan-Paul Lerch, Martin Wahl
TL;DR
This work analyzes empirical Hodge Laplacians as finite-sample approximations of the Laplace-Beltrami operator on differential forms under the manifold hypothesis. By constructing empirical $\ell$-forms and an empirical up-Hodge Laplacian, the authors derive a non-asymptotic, high-probability bound on the Dirichlet form error that closes the gap between discrete and continuous spectral energies. The analysis decomposes the error into a bias term (continuous Hodge-to-nonlocal approximation) and a variance term (concentration of U-statistics), and combines tools from exterior calculus, geometric analysis, and probabilistic concentration. The resulting bound improves existing Dirichlet-form convergence rates for graph Laplacians and provides a rigorous route toward spectral convergence results for higher-order Laplacians, with implications for Laplacian Eigenmaps, Diffusion Maps, and topological data analysis.
Abstract
Given i.i.d. observations uniformly distributed on a closed submanifold of the Euclidean space, we study higher-order generalizations of graph Laplacians, so-called Hodge Laplacians on graphs, as approximations of the Laplace-Beltrami operator on differential forms. Our main result is a high-probability error bound for the associated Dirichlet forms. This bound improves existing Dirichlet form error bounds for graph Laplacians in the context of Laplacian Eigenmaps, and it provides insights into the Betti numbers studied in topological data analysis and the complementing positive part of the spectrum.