  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.
