Sparse optimization with least-squares constraints

The use of convex optimization for the recovery of sparse signals from incomplete or compressed data is now common practice. Motivated by the success of basis pursuit in recovering sparse vectors, new formulations have been proposed that take advantage of different types of sparsity. In this paper we propose an efficient algorithm for solving a … Read more

Probing the Pareto frontier for basis pursuit solutions

The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously … Read more

Exact regularization of convex programs

The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a … Read more

Computing nonnegative tensor factorizations

Nonnegative tensor factorization (NTF) is a technique for computing a parts-based representation of high-dimensional data. NTF excels at exposing latent structures in datasets, and at finding good low-rank approximations to the data. We describe an approach for computing the NTF of a dataset that relies only on iterative linear-algebra techniques and that is comparable in … Read more

Exact regularization of linear programs

We show that linear programs (LPs) admit regularizations that either contract the original (primal) solution set or leave it unchanged. Any regularization function that is convex and has compact level sets is allowed–differentiability is not required. This is an extension of the result first described by Mangasarian and Meyer (SIAM J. Control Optim., 17(6), pp. … Read more