The game theoretic p-Laplacian and semi-supervised learning with few labels

TL;DR

This paper investigates the game theoretic p-Laplacian for semi-supervised learning on graphs, establishing its well-posedness and continuum limit, and analyzing the regularity of solutions using viscosity solutions and maximum principles.

Contribution

It introduces a rigorous analysis of the game theoretic p-Laplacian in semi-supervised learning, connecting discrete graph models to continuous p-Laplace equations.

Findings

The continuum limit of the graph-based semi-supervised learning with the p-Laplacian is a weighted p-Laplace equation.

Solutions to the graph p-Laplace equation are approximately Holder continuous with high probability.

The analysis employs viscosity solutions and maximum principles on graphs.

Abstract

We study the game theoretic p-Laplacian for semi-supervised learning on graphs, and show that it is well-posed in the limit of finite labeled data and infinite unlabeled data. In particular, we show that the continuum limit of graph-based semi-supervised learning with the game theoretic p-Laplacian is a weighted version of the continuous p-Laplace equation. We also prove that solutions to the graph p-Laplace equation are approximately Holder continuous with high probability. Our proof uses the viscosity solution machinery and the maximum principle on a graph.