  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.
