New Improved Penalty Methods for Sparse Reconstruction Based on Difference of Two Norms

Two new penalty methods for sparse reconstruction are proposed based on two types of difference of convex functions (DC for short) programming in which the DC objective functions are the difference of l1 and lσq norms and the difference of l1 and lr norms with r > 1. By introducing a generalized qterm shrinkage operator upon the special structure of lσq norm, we design a proximal gradient algorithm for handling the DC l1-lσq model. And by employing the majorization scheme, we develop a majorized penalty algorithm for the DC l1-lr model. The convergence results of our new algorithms are presented as well. Extensive simulation results show that these two new algorithms offer improved signal recovery performance and require reduced computational effort relative to state-of-the-art l1 and lp (p ∈ (0, 1))models.