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

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.

💡 Research Summary

This paper investigates the large‑sample asymptotics of Laplace learning—a semi‑supervised method that minimizes a graph‑based Dirichlet energy—when the data live in an infinite‑dimensional Hilbert space and are drawn from a Gaussian probability measure. In finite dimensions the continuum limit of the graph energy is well understood: after appropriate scaling by the ambient dimension (d), the discrete energy converges to a weighted Sobolev seminorm (\int |\nabla u(x)|^{2}\rho^{2}(x),dx), where (\rho) is the density of the data‑generating measure with respect to Lebesgue. In infinite dimensions there is no Lebesgue measure, so the density (\rho) does not exist and the classical scaling fails.

To overcome this obstacle the authors adopt a two‑fold strategy. First, they replace the Euclidean norm in the kernel with a fractional norm (|x|{X^{\alpha}}) defined through the eigenvalues ({\lambda_i}) of the covariance operator (C{\mu}) of the Gaussian measure:
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