Splitting Methods for Convex Clustering

TL;DR

This paper introduces two splitting algorithms, ADMM and AMA, for convex clustering, providing simple, unified frameworks that outperform previous methods in efficiency, and are adaptable to various norms.

Contribution

The paper presents novel ADMM and AMA algorithms for convex clustering, offering efficiency improvements and flexibility over existing methods.

Findings

AMA is significantly more efficient than ADMM.

Algorithms work well on simulated and real data.

Framework allows for new norms in convex clustering.

Abstract

Clustering is a fundamental problem in many scientific applications. Standard methods such as $k$-means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations of $k$-means and hierarchical clustering shrink cluster centroids toward one another and ensure a unique global minimizer. In this work we present two splitting methods for solving the convex clustering problem. The first is an instance of the alternating direction method of multipliers (ADMM); the second is an instance of the alternating minimization algorithm (AMA). In contrast to previously considered algorithms, our ADMM and AMA formulations provide simple and unified frameworks for solving the convex clustering problem under the previously studied norms and open the door to potentially novel norms. We demonstrate the performance of our algorithm on both simulated and real data examples. While the differences between the two algorithms appear to be minor on the surface, complexity analysis and numerical experiments show AMA to be significantly more efficient.