Reweighted ₁-Minimization for Sparse Solutions to Underdetermined Linear Systems

Numerical experiments have indicated that the reweighted ₁-minimization performs exceptionally well in locating sparse solutions of underdetermined linear systems of equations. We show that reweighted ₁-methods are intrinsically associated with the minimization of the so-called merit functions for sparsity, which are essentially concave approximations to the cardinality function. Based on this observation, we further show that a family of reweighted ₁-algorithms can be systematically derived from the perspective of concave optimization through the linearization technique. In order to conduct a unified convergence analysis for this family of algorithms, we introduce the concept of the range space property (RSP) of a matrix and prove that if its adjoint has this property, the reweighted ₁-algorithm can find a sparse solution to the underdetermined linear system provided that the merit function for sparsity is properly chosen. In particular, some convergence conditions for the Candès--Wakin--Boyd method and the recent ₚ-quasi-norm-based reweighted ₁-method can be obtained as special cases of the general framework.