Splitting Methods for Convex Clustering

• Published in: Journal of computational and graphical statistics Vol. 24; no. 4; pp. 994 - 1013
• Main Authors: Chi, Eric C., Lange, Kenneth
• Format: Journal Article
• Language: English
• Published: United States Taylor & Francis 02.10.2015; American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America; Taylor & Francis Ltd
• Subjects: Algorithms; Alternating direction method of multipliers; Alternating minimization algorithm; Cluster analysis; Clustering and Pattern Detection; Convex analysis; Convex optimization; Hierarchical clustering; k-means; Mathematical problems; Norms; Numerical analysis; Regularization paths; Studies; Regularization paths; k-means; Alternating direction method of multipliers; Hierarchical clustering; Convex optimization; Alternating minimization algorithm
• ISSN: 1061-8600; 1537-2715
• DOI: 10.1080/10618600.2014.948181

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. This article has supplementary materials available online.