Algorithms for Lipschitz Learning on Graphs

TL;DR

This paper introduces efficient algorithms for computing Lipschitz extensions on graphs, enabling fast regularization and interpolation of functions with minimal smoothness constraints.

Contribution

The authors develop novel algorithms for minimal and absolutely minimal Lipschitz extensions on graphs with improved expected linear and near-linear runtimes.

Findings

Expected linear time algorithm for minimal Lipschitz extension

Near-linear time algorithm for absolutely minimal Lipschitz extension

Efficient methods for $l_{0}$ and $l_{1}$ regularization on graph functions

Abstract

We develop fast algorithms for solving regression problems on graphs where one is given the value of a function at some vertices, and must find its smoothest possible extension to all vertices. The extension we compute is the absolutely minimal Lipschitz extension, and is the limit for large $p$ of $p$-Laplacian regularization. We present an algorithm that computes a minimal Lipschitz extension in expected linear time, and an algorithm that computes an absolutely minimal Lipschitz extension in expected time $\widetilde{O} (m n)$. The latter algorithm has variants that seem to run much faster in practice. These extensions are particularly amenable to regularization: we can perform $l_{0}$-regularization on the given values in polynomial time and $l_{1}$-regularization on the initial function values and on graph edge weights in time $\widetilde{O} (m^{3/2})$.