Accelerated Discrete Distribution Clustering under Wasserstein Distance

In a variety of research areas, the bag of weighted vectors and the histogram are widely used descriptors for complex objects. Both can be expressed as discrete distributions. D2-clustering pursues the minimum total within-cluster variation for a set of discrete distributions subject to the Kantorovich-Wasserstein metric. D2-clustering has a severe scalability issue, the bottleneck being the computation of a centroid distribution that minimizes its sum of squared distances to the cluster members. In this paper, we develop three scalable optimization techniques, specifically, the subgradient descent method, ADMM, and modified Bregman ADMM, for computing the centroids of large clusters without compromising the objective function. The strengths and weaknesses of these techniques are examined through experiments; and scenarios for their respective usage are recommended. Moreover, we develop both serial and parallelized versions of the algorithms, collectively named the AD2-clustering. By experimenting with large-scale data, we demonstrate the computational efficiency of the new methods and investigate their convergence properties and numerical stability. The clustering results obtained on several datasets in different domains are highly competitive in comparison with some widely used methods' in the corresponding areas.