The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
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| _version_ | 1866913234954485760 |
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| author | Holmgren, Benjamin Quist, Eli Schupbach, Jordan Fasy, Brittany Terese Rieck, Bastian |
| author_facet | Holmgren, Benjamin Quist, Eli Schupbach, Jordan Fasy, Brittany Terese Rieck, Bastian |
| contents | We introduce the manifold density function, which is an intrinsic method to validate manifold learning techniques. Our approach adapts and extends Ripley's $K$-function, and categorizes in an unsupervised setting the extent to which an output of a manifold learning algorithm captures the structure of a latent manifold. Our manifold density function generalizes to broad classes of Riemannian manifolds. In particular, we extend the manifold density function to general two-manifolds using the Gauss-Bonnet theorem, and demonstrate that the manifold density function for hypersurfaces is well approximated using the first Laplacian eigenvalue. We prove desirable convergence and robustness properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09529 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning Holmgren, Benjamin Quist, Eli Schupbach, Jordan Fasy, Brittany Terese Rieck, Bastian Machine Learning Algebraic Topology 57Z25 I.5.2 We introduce the manifold density function, which is an intrinsic method to validate manifold learning techniques. Our approach adapts and extends Ripley's $K$-function, and categorizes in an unsupervised setting the extent to which an output of a manifold learning algorithm captures the structure of a latent manifold. Our manifold density function generalizes to broad classes of Riemannian manifolds. In particular, we extend the manifold density function to general two-manifolds using the Gauss-Bonnet theorem, and demonstrate that the manifold density function for hypersurfaces is well approximated using the first Laplacian eigenvalue. We prove desirable convergence and robustness properties. |
| title | The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning |
| topic | Machine Learning Algebraic Topology 57Z25 I.5.2 |
| url | https://arxiv.org/abs/2402.09529 |