Hodge Laplacians and Hodge Diffusion Maps
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913787648409600 |
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| author | Gomez, Alvaro Almeida Franco, Jorge Duque |
| author_facet | Gomez, Alvaro Almeida Franco, Jorge Duque |
| contents | We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms, thereby providing an approximation of the Hodge Laplacian operator. Hodge Diffusion Maps extend existing non-linear dimensionality reduction techniques, including vector diffusion maps, as well as the theories behind diffusion maps and Laplacian Eigenmaps. Our approach captures higher-order topological features of the data-set by projecting it into lower-dimensional Euclidean spaces using the Hodge Laplacian. We develop a theoretical framework to estimate the approximation error of the exterior derivative, based on sample points distributed over a real manifold. Numerical experiments support and validate the proposed methodology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07910 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge Laplacians and Hodge Diffusion Maps Gomez, Alvaro Almeida Franco, Jorge Duque Machine Learning 68P05, 68T10, 68T45, 68W25 We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms, thereby providing an approximation of the Hodge Laplacian operator. Hodge Diffusion Maps extend existing non-linear dimensionality reduction techniques, including vector diffusion maps, as well as the theories behind diffusion maps and Laplacian Eigenmaps. Our approach captures higher-order topological features of the data-set by projecting it into lower-dimensional Euclidean spaces using the Hodge Laplacian. We develop a theoretical framework to estimate the approximation error of the exterior derivative, based on sample points distributed over a real manifold. Numerical experiments support and validate the proposed methodology. |
| title | Hodge Laplacians and Hodge Diffusion Maps |
| topic | Machine Learning 68P05, 68T10, 68T45, 68W25 |
| url | https://arxiv.org/abs/2504.07910 |