Continuum Limit of Lipschitz Learning on Graphs

TL;DR

This paper establishes the continuum limit of Lipschitz learning on graphs by proving $ ext{Gamma}$-convergence of functionals approximating the Lipschitz constant, extending understanding of graph-based semi-supervised learning methods.

Contribution

It provides the first rigorous $ ext{Gamma}$-convergence proof for Lipschitz learning on graphs, connecting discrete graph functionals to continuum PDE limits.

Findings

Proves $ ext{Gamma}$-convergence of Lipschitz learning functionals to the supremum norm of the gradient.

Shows convergence of minimizers and graph distance functions to geodesic distances.

Allows for varying labeled data sets converging to a closed set in Hausdorff distance.

Abstract

Tackling semi-supervised learning problems with graph-based methods has become a trend in recent years since graphs can represent all kinds of data and provide a suitable framework for studying continuum limits, e.g., of differential operators. A popular strategy here is $p$-Laplacian learning, which poses a smoothness condition on the sought inference function on the set of unlabeled data. For $p<\infty$ continuum limits of this approach were studied using tools from $\Gamma$-convergence. For the case $p=\infty$, which is referred to as Lipschitz learning, continuum limits of the related infinity-Laplacian equation were studied using the concept of viscosity solutions. In this work, we prove continuum limits of Lipschitz learning using $\Gamma$-convergence. In particular, we define a sequence of functionals which approximate the largest local Lipschitz constant of a graph function and prove $\Gamma$-convergence in the $L^\infty$-topology to the supremum norm of the gradient as the graph becomes denser. Furthermore, we show compactness of the functionals which implies convergence of minimizers. In our analysis we allow a varying set of labeled data which converges to a general closed set in the Hausdorff distance. We apply our results to nonlinear ground states, i.e., minimizers with constrained $L^p$-norm, and, as a by-product, prove convergence of graph distance functions to geodesic distance functions.