Consistency of Lipschitz learning with infinite unlabeled data and finite labeled data

TL;DR

This paper investigates the behavior of Lipschitz learning on graphs with infinite unlabeled and finite labeled data, revealing conditions under which it is sensitive or insensitive to data distribution.

Contribution

It proves the conjecture of insensitivity in a specific graph model and demonstrates how to control sensitivity through weight tuning in a more general setting.

Findings

Lipschitz learning converges to an $ty$-Laplace type equation.

In a random geometric graph with kernel weights, Lipschitz learning is insensitive to data distribution.

In a self-tuning weight graph, sensitivity to data distribution can be adjusted.

Abstract

We study the consistency of Lipschitz learning on graphs in the limit of infinite unlabeled data and finite labeled data. Previous work has conjectured that Lipschitz learning is well-posed in this limit, but is insensitive to the distribution of the unlabeled data, which is undesirable for semi-supervised learning. We first prove that this conjecture is true in the special case of a random geometric graph model with kernel-based weights. Then we go on to show that on a random geometric graph with self-tuning weights, Lipschitz learning is in fact highly sensitive to the distribution of the unlabeled data, and we show how the degree of sensitivity can be adjusted by tuning the weights. In both cases, our results follow from showing that the sequence of learned functions converges to the viscosity solution of an $\infty$-Laplace type equation, and studying the structure of the limiting equation.