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# Algorithms for Lipschitz Learning on Graphs

*Proceedings of The 28th Conference on Learning Theory*, PMLR 40:1190-1223, 2015.

#### 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 \widetildeO (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 \widetildeO (m^3/2). Our definitions and algorithms naturally extend to directed graphs.