After reviewing the use of l_1-minimization for the recovery of sparse vectors, we investigate the advantages and drawbacks of substituting it by an l_q-minimization for 0 < q < 1. On the theoretical side, we see that the Restricted Isometry Condition guaranteeing recovery becomes weaker and that the class of suitable random matrices becomes larger. On the algorithmic side, we introduce an iteratively reweighted l_1-minimization scheme to approximate the nonconvex l_q-minimization and we discuss its positive and negative features.
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