Pursuit of the Cluster Structure of Network Lasso: Recovery Condition and Non-convex Extension

TL;DR

This paper analyzes the cluster recovery conditions of Network Lasso and introduces a non-convex extension called Network Trimmed Lasso, which improves cluster detection and provides an efficient algorithm with convergence guarantees.

Contribution

It establishes sufficient conditions for cluster recovery in Network Lasso and develops a non-convex extension with an ADMM algorithm, enhancing clustering performance.

Findings

Non-convex NTL better reveals cluster structure when NL fails.

ADMM algorithm effectively solves NTL with proven convergence.

Numerical results show improved clustering clarity with NTL.

Abstract

Network Lasso (NL for short) is a methodology for estimating models by simultaneously clustering data samples and fitting the models to the samples. It often succeeds in forming clusters thanks to the geometry of the $\ell_1$-regularizer employed therein, but there might be limitations because of the convexity of the regularizer. This paper focuses on the cluster structure that NL yields and reinforces it by developing a non-convex extension, which we call Network Trimmed Lasso (NTL for short). Specifically, we first study a sufficient condition that guarantees the recovery of the latent cluster structure of NL on the basis of the result of Sun et al. (2021) for Convex Clustering, which is a special case of NL for clustering. Second, we extend NL to NTL to incorporate a cardinality (or, $\ell_0$-)constraint and rewrite the constrained optimization problem defined with the $\ell_0$ norm, a discontinuous function, into an equivalent unconstrained continuous optimization problem. We develop ADMM algorithms to solve NTL and provide its convergence results. Numerical illustrations demonstrate that the non-convex extension provides a more clear-cut cluster structure when NL fails to form clusters without incorporating prior knowledge of the associated parameters.