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On empirical Hodge Laplacians under the manifold hypothesis
[Submitted on 4 Apr 2025 (v1), last revised 17 Jul 2026 (this ve · 2025-04-04 · via math updates on arXiv.org

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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.

Submission history

From: Martin Wahl [view email]
[v1] Fri, 4 Apr 2025 13:18:32 UTC (30 KB)
[v2] Fri, 17 Jul 2026 09:01:18 UTC (3,539 KB)