On empirical Hodge Laplacians under the manifold hypothesis
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866909565205872640 |
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| author | Lerch, Jan-Paul Wahl, Martin |
| author_facet | Lerch, Jan-Paul Wahl, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On empirical Hodge Laplacians under the manifold hypothesis Lerch, Jan-Paul Wahl, Martin Statistics Theory Differential Geometry Probability 62R30, 62R40, 53Z50, 35K08, 05C80 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. |
| title | On empirical Hodge Laplacians under the manifold hypothesis |
| topic | Statistics Theory Differential Geometry Probability 62R30, 62R40, 53Z50, 35K08, 05C80 |
| url | https://arxiv.org/abs/2504.03427 |