  The non-convex $\alpha\|\cdot\|_{\ell_1}-\beta\| \cdot\|_{\ell_2}$
$(\alpha\ge\beta\geq0)$ regularization has attracted attention in the field of
sparse recovery. One way to obtain a minimizer of this regularization is the
ST-($\alpha\ell_1-\beta\ell_2$) 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-($\alpha\ell_1-\beta\ell_2$) algorithm by extending the
PG method to the non-convex $\alpha\ell_1-\beta\ell_2$ 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.
