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Recovery of Sparse Probability Measures via Convex Programming. presented at
 Advances in Neural Information Processing Systems; 2012. (unpublished
"... We consider the problem of cardinality penalized optimization of a convex function over the probability simplex with additional convex constraints. The classical `1 regularizer fails to promote sparsity on the probability simplex since `1 norm on the probability simplex is trivially constant. We pr ..."
Abstract

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We consider the problem of cardinality penalized optimization of a convex function over the probability simplex with additional convex constraints. The classical `1 regularizer fails to promote sparsity on the probability simplex since `1 norm on the probability simplex is trivially constant. We propose a direct relaxation of the minimum cardinality problem and show that it can be efficiently solved using convex programming. As a first application we consider recovering a sparse probability measure given moment constraints, in which our formulation becomes linear programming, hence can be solved very efficiently. A sufficient condition for exact recovery of the minimum cardinality solution is derived for arbitrary affine constraints. We then develop a penalized version for the noisy setting which can be solved using second order cone programs. The proposed method outperforms known rescaling heuristics based on `1 norm. As a second application we consider convex clustering using a sparse Gaussian mixture and compare our results with the well known soft kmeans algorithm. 1