Nonconvex Sorted $\ell_1$ Minimization for Sparse Approximation

The $\ell_1$ norm is the tight convex relaxation for the $\ell_0$ "norm" and has been successfully applied for recovering sparse signals. For problems with fewer samplings, one needs to enhance the sparsity by nonconvex penalties such as $\ell_p$ "norm". As one method for solving $\ell_p$ minimization problems, iteratively reweighted $\ell_1$ minimization updates the weight for each component based on the value of the same component at the previous iteration. It assigns large weights on small components in magnitude and small weights on large components in magnitude. In this paper, we consider a weighted $\ell_1$ penalty with the set of the weights fixed and the weights are assigned based on the sort of all the components in magnitude. The smallest weight is assigned to the largest component in magnitude. This new penalty is called nonconvex sorted $\ell_1$. Then we propose two methods for solving nonconvex sorted $\ell_1$ minimization problems: iteratively reweighted $\ell_1$ minimization and iterative sorted thresholding, and prove that both methods will converge to a local optimum. We also show that both methods are generalizations of iterative support detection and iterative hard thresholding respectively. The numerical experiments demonstrate the better performance of assigning weights by sort compared to $\ell_p$ minimization.