Sparsest solutions of underdetermined linear systems via lq-minimization for 0<q≤1

Article history: Received 14 December 2007 Revised 9 September 2008 Accepted 11 September 2008 Available online 25 September 2008 Communicated by Naoki Saito We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal q-quasinorm is also the sparsest one. This generalizes, and slightly improves, a similar result for the 1-norm. We then introduce a simple numerical scheme to compute solutions with minimal q-quasinorm, and we study its convergence. Finally, we display the results of some experiments which indicate that the q-method performs better than other available methods. © 2008 Elsevier Inc. All rights reserved.