Dissimilarity-based Sparse Subset Selection
  Finding an informative subset of a large collection of data points or models is at the center of many problems in computer vision, recommender systems, bio/health informatics as well as image and natural language processing. Given pairwise dissimilarities between the elements of a `source set' and a `target set,' we consider the problem of finding a subset of the source set, called representatives or exemplars, that can efficiently describe the target set. We formulate the problem as a row-sparsity regularized trace minimization problem. Since the proposed formulation is, in general, NP-hard, we consider a convex relaxation. The solution of our optimization finds representatives and the assignment of each element of the target set to each representative, hence, obtaining a clustering. We analyze the solution of our proposed optimization as a function of the regularization parameter. We show that when the two sets jointly partition into multiple groups, our algorithm finds representatives from all groups and reveals clustering of the sets. In addition, we show that the proposed framework can effectively deal with outliers. Our algorithm works with arbitrary dissimilarities, which can be asymmetric or violate the triangle inequality. To efficiently implement our algorithm, we consider an Alternating Direction Method of Multipliers (ADMM) framework, which results in quadratic complexity in the problem size. We show that the ADMM implementation allows to parallelize the algorithm, hence further reducing the computational time. Finally, by experiments on real-world datasets, we show that our proposed algorithm improves the state of the art on the two problems of scene categorization using representative images and time-series modeling and segmentation using representative~models. 
Ehsan Elhamifar, Guillermo Sapiro and S. Shankar Sastry
cs.LG stat.ML