Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions

• URL: http://arxiv.org/abs/2601.14515v1
• Date: Tue, 20 Jan 2026 22:14:05 GMT
• Title: Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions
• Authors: Zhengang Zhong, Yury Korolev, Matthew Thorpe,
• Abstract summary: Laplace learning is a solution for finding missing labels from a partially labeled dataset.<n>The Lebesgue measure on infinite-dimensional spaces requires the analysis if the data aren't finite-dimensional.
• Score: 2.020917258669917
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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