Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908779044405248 |
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| author | Zhong, Zhengang Korolev, Yury Thorpe, Matthew |
| author_facet | Zhong, Zhengang Korolev, Yury Thorpe, Matthew |
| contents | Laplace learning is a semi-supervised method, a solution for finding missing labels from a partially labeled dataset utilizing the geometry given by the unlabeled data points. The method minimizes a Dirichlet energy defined on a (discrete) graph constructed from the full dataset. In finite dimensions the asymptotics in the large (unlabeled) data limit are well understood with convergence from the graph setting to a continuum Sobolev semi-norm weighted by the Lebesgue density of the data-generating measure. The lack of the Lebesgue measure on infinite-dimensional spaces requires rethinking the analysis if the data aren't finite-dimensional. In this paper we make a first step in this direction by analyzing the setting when the data are generated by a Gaussian measure on a Hilbert space and proving pointwise convergence of the graph Dirichlet energy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14515 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions Zhong, Zhengang Korolev, Yury Thorpe, Matthew Machine Learning Numerical Analysis Laplace learning is a semi-supervised method, a solution for finding missing labels from a partially labeled dataset utilizing the geometry given by the unlabeled data points. The method minimizes a Dirichlet energy defined on a (discrete) graph constructed from the full dataset. In finite dimensions the asymptotics in the large (unlabeled) data limit are well understood with convergence from the graph setting to a continuum Sobolev semi-norm weighted by the Lebesgue density of the data-generating measure. The lack of the Lebesgue measure on infinite-dimensional spaces requires rethinking the analysis if the data aren't finite-dimensional. In this paper we make a first step in this direction by analyzing the setting when the data are generated by a Gaussian measure on a Hilbert space and proving pointwise convergence of the graph Dirichlet energy. |
| title | Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2601.14515 |