Reweighted $\ell_1$-Minimization for Sparse Solutions to Underdetermined Linear Systems

Abstract

Numerical experiments have indicated that the reweighted $\ell_1$-minimization performs exceptionally well in locating sparse solutions of underdetermined linear systems of equations. We show that reweighted $\ell_1$-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 $\ell_1$-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 $\ell_1$-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 Can...