A projected gradient method for $α\ell_{1}-β\ell_{2}$ sparsity regularization

Abstract

The non-convex $α\|\cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ $(α\geβ\geq0)$ regularization has attracted attention in the field of sparse recovery. One way to obtain a minimizer of this regularization is the ST-($α\ell_1-β\ell_2$) 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 the classical $\ell_1$ sparsity regularization. In this paper, we present two accelerated alternatives to the ST-($α\ell_1-β\ell_2$) algorithm by extending the PG method to the non-convex $α\ell_1-β\ell_2$ sparsity regularization. Moreover, we discuss a strategy to determine the radius $R$ of the $\ell_1$-ball constraint by Morozov's discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach.