Computational Aspects of Constrained L 1-L 2 Minimization for Compressive Sensing

We study the computational properties of solving a constrained L 1-L 2 minimization via a difference of convex algorithm (DCA), which was proposed in our previous work [13,19] to recover sparse signals from a under-determined linear system. We prove that the DCA converges to a stationary point of the nonconvex L 1-L 2 model. We clarify the relationship of DCA to a convex method, Bregman iteration [20] for solving a constrained L 1 minimization. Through experiments, we discover that both L 1 and L 1-L 2 obtain better recovery results from more coherent matrices, which appears unknown in theoretical analysis of exact sparse recovery. In addition, numerical studies motivate us to consider a weighted difference model L 1-αL 2 (α > 1) to deal with ill-conditioned matrices when L 1-L 2 fails to obtain a good solution.