In a variety of research areas, the weighted bag of 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, called Wasserstein barycenter, that minimizes its sum of squared distances to the cluster members. In this paper, we develop a modified Bregman ADMM approach for computing the approximate discrete Wasserstein barycenter of large clusters. In the case when the support points of the barycenters are unknown and have low cardinality, our method achieves high accuracy empirically at a much reduced computational cost. The strengths and weaknesses of our method and its alternatives are examined through experiments, and we recommend scenarios for their respective usage. Moreover, we develop both serial and parallelized versions of the algorithm. 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.
@cite_7: We present new algorithms to compute the mean of a set of empirical probability measures under the optimal transport metric. This mean, known as the Wasserstein barycenter, is the measure that minimizes the sum of its Wasserstein distances to each element in that set. We propose two original algorithms to compute Wasserstein barycenters that build upon the subgradient method. A direct implementation of these algorithms is, however, too costly because it would require the repeated resolution of large primal and dual optimal transport problems to compute subgradients. Extending the work of Cuturi (2013), we propose to smooth the Wasserstein distance used in the definition of Wasserstein barycenters with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms. We use these algorithms to visualize a large family of images and to solve a constrained clustering problem.
@cite_6: Developing effective methods for automated annotation of digital pictures continues to challenge computer scientists. The capability of annotating pictures by computers can lead to breakthroughs in a wide range of applications, including Web image search, online picture-sharing communities, and scientific experiments. In this work, the authors developed new optimization and estimation techniques to address two fundamental problems in machine learning. These new techniques serve as the basis for the automatic linguistic indexing of pictures - real time (ALIPR) system of fully automatic and high-speed annotation for online pictures. In particular, the D2-clustering method, in the same spirit as K-Means for vectors, is developed to group objects represented by bags of weighted vectors. Moreover, a generalized mixture modeling technique (kernel smoothing as a special case) for nonvector data is developed using the novel concept of hypothetical local mapping (HLM). ALIPR has been tested by thousands of pictures from an Internet photo-sharing site, unrelated to the source of those pictures used in the training process. Its performance has also been studied at an online demonstration site, where arbitrary users provide pictures of their choices and indicate the correctness of each annotation word. The experimental results show that a single computer processor can suggest annotation terms in real time and with good accuracy.
In applications on high-dimensional data, it is desirable to optimize the support points rather than fix them from the beginning. This however leads to a non-convex optimization problem. Our work aims at developing practical numerical methods. In particular, the method optimizes jointly the locations and weights of the support points in a single loop without resorting to a bi-level optimization reformulation, as was done in earlier work @cite_6 @cite_7 .