${\ell }_1^2-\hat{I}\cdot {\ell }_2^2$ regularization for sparse recovery

Abstract

This paper presents a regularization technique incorporating a non-convex and non-smooth term, ${\ell }_1^2-\hat{I}\cdot {\ell }_2^2$, with parameters $0<\hat{I}\cdot \le 1$ designed to address ill-posed linear problems that yield sparse solutions. We explore the existence, stability, and convergence of the regularized solution, demonstrating that the ${\ell }_1^2-\hat{I}\cdot {\ell }_2^2$ regularization is well-posed and results in sparse solutions. Under suitable source conditions, we establish a convergence rate of $\mathcal{O}\left(\hat{I}´\right)$ in the ${\ell }_2$-norm for both a priori and a posteriori parameter choice rules. Additionally, we propose and analyze a numerical algorithm based on a half-variation iterative strategy combined with the proximal gradient method. We prove convergence despite the regularization term being non-smooth and non-convex. The algorithm features a straightforward structure, facilitating implementation. Furthermore, we propose a projected gradient iterative strategy base on surrogate function approach to achieve faster solving. Experimentally, we demonstrate visible improvements of ${\ell }_1^2-\hat{I}\cdot {\ell }_2^2$ over ${\ell }_1$, ${\ell }_1-\hat{I}\cdot {\ell }_2$, and other nonconvex regularizations for compressive sensing and image deblurring problems. All the numerical results show the efficiency of our proposed approach.

40 pages, 9 figures