Splitting Methods for Convex Clustering
This work addresses the issue of suboptimal local minima in clustering for scientific applications, but it is incremental as it builds on existing convex relaxations with new algorithmic formulations.
The authors tackled the problem of local minima in clustering by introducing two splitting methods, ADMM and AMA, to solve convex relaxations of k-means and hierarchical clustering, with AMA shown to be significantly more efficient in complexity analysis and numerical experiments.
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.