Exact Recovery Conditions For Sparse Signal

In this paper, we utilize two classes of relaxations-convex relax-ation and nonconvex relaxation-to study the exact recovery conditions for sparse signal. Convex models contain the standard l1minimization and weighted l1min-imization, the later of which has exceptional numerical performance, while the nonconvex relaxation mainly is centered on lq minimization. We concentrate on doing research on weighted l1and(0<q<1) minimizations. For the former model, we first introduce the concept of WNSP (weighted null space property) and reveal that it is a necessary and sufficient condition for exact recovery. The RIC bounds related to restrict isometry property (RIP) have been given for both models and also to be used to ensure the exact recovery. We will present that those bounds we obtained are much better than current results under some mild cases. In addition, we achieve a modified iterative reweighted l1minimization (MIRL1) al-gorithm based on our weighted thory, and the numerical experiments demonstrate that our algorithm behaves much better than non-weighted l1minimization.