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Compressed sensing is a new methodology for constructing sensors which allowsparse signals to be efficiently recovered using only a small number ofobservations. The recovery problem can often be stated as the one of findingthe solution of an underdetermined system of linear equations with the smallestpossible support. The most studied relaxation of this hard combinatorialproblem is the $l_1$-relaxation consisting of searching for solutions withsmallest $l_1$-norm. In this short note, based on the ideas of Lagrangianduality, we introduce an alternating $l_1$ relaxation for the recovery problemenjoying higher recovery rates in practice than the plain $l_1$ relaxation andthe recent reweighted $l_1$ method of Cand\`es, Wakin and Boyd.

An Alternating l1 approach to the compressed sensing problem