Nonconvex Sorted l1 Minimization for Sparse Approximation

The l1 norm is the tight convex relaxation for the l0 “norm” and has been successfully applied for recovering sparse signals. However, for problems with fewer samples than required for accurate l1 recovery, one needs to apply nonconvex penalties such as lp “norm”. As one method for solving lp minimization problems, iteratively reweighted l1 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. The set of the weights is not fixed, and it makes the analysis of this method difficult. In this paper, we consider a weighted l1 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 l1. Then we propose two methods for solving nonconvex sorted l1 minimization problems: iteratively reweighted l1 minimization and iterative sorted thresholding, and prove that both methods will converge to a local minimizer of the nonconvex sorted l1 minimization problems. We also show that both methods are generalizations of iterative This work is partially supported by ERC AdG A-DATADRIVE-B, IUAP-DYSCO, GOAMANET and OPTEC, National Natural Science Foundation of China (11201079), the Fundamental Research Funds for the Central Universities of China (20520133238, 20520131169), and NSF grants DMS-0748839 and DMS-1317602. X. Huang Department of Electrical Engineering, KU Leuven, B-3001 Leuven, Belgium. E-mail: [email protected] L. Shi Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, 200433, P.R. China. E-mail: [email protected] M. Yan Department of Mathematics, University of California, Los Angeles, CA 90095, USA. E-mail: [email protected] 2 Xiaolin Huang et al. support detection and iterative hard thresholding respectively. The numerical experiments demonstrate the better performance of assigning weights by sort compared to assigning by value.