Exact Recovery for Sparse Signal via Weighted l_1 Minimization

Numerical experiments in literature on compressed sensing have indicated that the reweighted l1 minimization performs exceptionally well in recovering sparse signal. In this paper, we develop exact recovery conditions and algorithm for sparse signal via weighted l1 minimization from the insight of the classical NSP (null space property) and RIC (restricted isometry constant) bound. We first introduce the concept of WNSP (weighted null space property) and reveal that it is a necessary and sufficient condition for exact recovery. We then prove that the RIC bound by weighted l1 minimization is δak < √ a−1 a−1+γ2 , where a > 1, 0< γ≤ 1 is determined by an optimization problem over the null space. When γ< 1 this bound is greater than √ a−1 a from l1 minimization. In addition, we also establish the bound on δk and show that it can be larger than the sharp one 1/3 via l1 minimization and also greater than 0.4343 via weighted l1 minimization under some mild cases. Finally, we achieve a modified iterative reweighted l1 minimization (MIRL1) algorithm based on our selection principle of weight, and the numerical experiments demonstrate that our algorithm behaves much better than l1 minimization and iterative reweighted l1 minimization (IRL1) algorithm.