An Alternating l1 approach to the compressed sensing problem

Compressed sensing is a new methodology for constructing sensors which allow sparse signals to be efficiently recovered using only a small number of observations. The recovery problem can often be stated as the one of finding the solution of an underdetermined system of linear equations with the smallest possible support. The most studied relaxation of this hard combinatorial problem is the l1-relaxation consisting of searching for solutions with smallest l1-norm. In this short note, based on the ideas of Lagrangian duality, we introduce an alternating l1 relaxation for the recovery problem enjoying higher recovery rates in practice than the plain l1 relaxation and the recent reweighted l1 method of Candès, Wakin and Boyd.