Uniform Convergence Rates for Lipschitz Learning on Graphs

TL;DR

This paper establishes uniform convergence rates for solutions of the graph infinity Laplace equation in Lipschitz learning, even on sparsely connected graphs, bridging discrete graph solutions with continuum Lipschitz extensions.

Contribution

It provides the first quantitative convergence rates for Lipschitz learning solutions on very sparse graphs, including those with minimal connectivity.

Findings

Convergence rates hold for graphs with connectivity radius down to the minimal level.

Graph distance functions converge quantitatively to geodesic distances.

The framework applies under very general assumptions on graph weights and sampling.

Abstract

Lipschitz learning is a graph-based semi-supervised learning method where one extends labels from a labeled to an unlabeled data set by solving the infinity Laplace equation on a weighted graph. In this work we prove uniform convergence rates for solutions of the graph infinity Laplace equation as the number of vertices grows to infinity. Their continuum limits are absolutely minimizing Lipschitz extensions with respect to the geodesic metric of the domain where the graph vertices are sampled from. We work under very general assumptions on the graph weights, the set of labeled vertices, and the continuum domain. Our main contribution is that we obtain quantitative convergence rates even for very sparsely connected graphs, as they typically appear in applications like semi-supervised learning. In particular, our framework allows for graph bandwidths down to the connectivity radius. For proving this we first show a quantitative convergence statement for graph distance functions to geodesic distance functions in the continuum. Using the "comparison with distance functions" principle, we can pass these convergence statements to infinity harmonic functions and absolutely minimizing Lipschitz extensions.