A two-term penalty function for inverse problems with sparsity constrains

Inverse problems with sparsity constrains, such Basis Pursuit denoising (BPDN) and Convolutional BPDN (CBPDN), usually use the '1-norm as the penalty function; however such choice leads to a solution that is biased towards zero. Recently, several works have proposed and assessed the properties of other non-standard penalty functions (most of them non-convex), which avoid the above mentioned drawback and at the same time are intended to induce sparsity more strongly than the '1-norm. In this paper we propose a two-term penalty function consisting of a synthesis between the '1-norm and the penalty function associated with the Non-Negative Garrote (NNG) thresholding rule. Although the proposed two-term penalty function is nonconvex, the total cost function for the BPDN/CBPDN problems is still convex. The performance of the proposed twoterm penalty function is compared with other reported choices for practical denoising, deconvolution and convolutional sparse coding (CSC) problems within the BPDN/CBPDN frameworks. Our experimental results show that the proposed two-term penalty function is particularly effective (better reconstruction with sparser solutions) for the CSC problem while attaining competitive performance for the denoising and deconvolution problems.