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 ap...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 support detection and iterative hard thresholding,respectively.The numerical experiments demonstrate the better performance of assigning weights by sort compared to assigning by value.展开更多
基金This work is partially supported by European Research Council,the National Natural Science Foundation of China(No.11201079)the Fundamental Research Funds for the Central Universities of China(Nos.20520133238 and 20520131169)the National Natural Science Foundation of United States(Nos.DMS-0748839 and DMS-1317602).
文摘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 support detection and iterative hard thresholding,respectively.The numerical experiments demonstrate the better performance of assigning weights by sort compared to assigning by value.
