A projected gradient method for $\alpha\ell_{1}-\beta\ell_{2}$ sparsity regularization

TL;DR

This paper introduces accelerated projected gradient methods for non-convex $\,\alpha\ell_1-\beta\ell_2$ sparsity regularization, improving convergence speed over traditional algorithms in sparse recovery tasks.

Contribution

It extends the projected gradient method to non-convex $\,\alpha\ell_1-\beta\ell_2$ regularization and proposes two accelerated algorithms with a strategy for setting the constraint radius.

Findings

Accelerated methods outperform classical algorithms in convergence speed.

The proposed approach effectively handles non-convex sparsity regularization.

Numerical experiments demonstrate the efficiency of the methods.

Abstract

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.