Nonconvex Sorted ℓ₁ Minimization for Sparse Approximation

الأصل · EN

The ℓ₁ norm is the tight convex relaxation for the ℓ₀ "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 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. 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 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 ℓₚ minimization.