A projected gradient method for αℓ1-βℓ2 sparsity regularization

Abstract

The non-convex α∥ ∥ℓ1 - β∥ ∥ℓ2 (α ∥ β ∥ 0) regularization is a new approach for sparse recovery.Aminimizer of the α∥ ∥ℓ1 - β∥ ∥ℓ2 regularized function can be computed by applying the ST-(αℓ1 - βℓ2) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to solve problems with the classical ℓ1 sparsity regularization. In this paper, we present two accelerated alternatives to the ST-(αℓ1 - βℓ2) algorithmby extending the PG method to the non-convexα∥ ∥ℓ1 - β∥ ∥ℓ2 sparsity regularization. Moreover, we discuss a strategy to determine the radius R of the ℓ1-ball constraint by Morozov's discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach.