Large Data Limits of Laplace Learning for Gaussian Measure Data in Infinite Dimensions
This work addresses a foundational gap in semi-supervised learning theory for high-dimensional data, though it is incremental as it builds on existing finite-dimensional results.
The paper tackles the problem of extending Laplace learning's large data limit analysis to infinite-dimensional spaces, where traditional Lebesgue measures do not apply, by proving pointwise convergence of the graph Dirichlet energy for data generated by a Gaussian measure in a Hilbert space.
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.