A projected gradient method for $\alpha {\ell }_1-\beta {\ell }_2$ sparsity regularization

Abstract

The non-convex $\alpha \Vert \cdot {\Vert }_{{\ell }_1}-\beta \Vert \cdot {\Vert }_{{\ell }_2}$ $(\alpha \ge \beta \ge 0)$ 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.

arXiv comment30 pages; 8 figures

References in corpus (5)

• NESTA: A Fast and Accurate First-order Method for Sparse Recovery
• Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints
• On a generalization of the iterative soft-thresholding algorithm for the case of non-separable penalty
• Accelerating gradient projection methods for ${\ell }_1$-constrained signal recovery by steplength selection rules
• Nonconvex Sorted ${\ell }_1$ Minimization for Sparse Approximation