On the Duality between Network Flows and Network Lasso

TL;DR

This paper reveals a precise duality between network Lasso and network flow problems, linking machine learning on networked data with classical graph algorithms like flow and clustering.

Contribution

It establishes a formal equivalence between network Lasso solutions and minimum-cost network flows, providing new insights into their relationship.

Findings

nLasso is equivalent to a minimum-cost flow problem on the data network

Characterization of nLasso solutions via network flows

Link between nLasso, clustering, and maximum flow algorithms

Abstract

Many applications generate data with an intrinsic network structure such as time series data, image data or social network data. The network Lasso (nLasso) has been proposed recently as a method for joint clustering and optimization of machine learning models for networked data. The nLasso extends the Lasso from sparse linear models to clustered graph signals. This paper explores the duality of nLasso and network flow optimization. We show that, in a very precise sense, nLasso is equivalent to a minimum-cost flow problem on the data network structure. Our main technical result is a concise characterization of nLasso solutions via existence of certain network flows. The main conceptual result is a useful link between nLasso methods and basic graph algorithms such as clustering or maximum flow.