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

TL;DR

This paper investigates the limits of Laplace learning in infinite-dimensional spaces, specifically with Gaussian measure data, by analyzing the convergence of graph Dirichlet energy in such settings.

Contribution

It provides the first analysis of Laplace learning's asymptotics for Gaussian measure data in infinite-dimensional Hilbert spaces, extending finite-dimensional understanding.

Findings

Proves pointwise convergence of graph Dirichlet energy in infinite dimensions.

Extends Laplace learning analysis to Gaussian measures on Hilbert spaces.

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.