title: Splitting Methods for Convex Clustering.
abstract: Clustering is a fundamental problem in many scientific applications. Standard methods such ask-means, Gaussian mixture models, and hierarchical clustering, however, are beset by local minima, which are sometimes drastically suboptimal. Recently introduced convex relaxations ofk-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. This article has supplemental materials available online.
authors.affiliation: Postdoctoral Research Associate, Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005.
authors.forename: Eric C
authors.initials: EC
authors.lastname: Chi
authors.affiliation: Professor, Departments of Biomathematics, Human Genetics, and Statistics, University of California, Los Angeles, CA 90095-7088.
authors.forename: Kenneth
authors.initials: K
authors.lastname: Lange
keywords: Alternating direction method of multipliers; Alternating minimization algorithm; Convex optimization; Hierarchical clustering; Regularization paths; k-means