$\ell_{1}^{2}-η\ell_{2}^{2}$ regularization for sparse recovery
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912427480711168 |
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| author | Li, Long Ding, Liang |
| author_facet | Li, Long Ding, Liang |
| contents | This paper presents a regularization technique incorporating a non-convex and non-smooth term, $\ell_{1}^{2}-η\ell_{2}^{2}$, with parameters $0<η\leq 1$ 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 $\ell_{1}^{2}-η\ell_{2}^{2}$ regularization is well-posed and results in sparse solutions. Under suitable source conditions, we establish a convergence rate of $\mathcal{O}\left(δ\right)$ 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 $\ell_{1}^{2}-η\ell_{2}^{2}$ over $\ell_{1}$, $\ell_{1}-η\ell_{2}$, and other nonconvex regularizations for compressive sensing and image deblurring problems. All the numerical results show the efficiency of our proposed approach. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_11372 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\ell_{1}^{2}-η\ell_{2}^{2}$ regularization for sparse recovery Li, Long Ding, Liang Optimization and Control 47A52 G.1.6 This paper presents a regularization technique incorporating a non-convex and non-smooth term, $\ell_{1}^{2}-η\ell_{2}^{2}$, with parameters $0<η\leq 1$ 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 $\ell_{1}^{2}-η\ell_{2}^{2}$ regularization is well-posed and results in sparse solutions. Under suitable source conditions, we establish a convergence rate of $\mathcal{O}\left(δ\right)$ 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 $\ell_{1}^{2}-η\ell_{2}^{2}$ over $\ell_{1}$, $\ell_{1}-η\ell_{2}$, and other nonconvex regularizations for compressive sensing and image deblurring problems. All the numerical results show the efficiency of our proposed approach. |
| title | $\ell_{1}^{2}-η\ell_{2}^{2}$ regularization for sparse recovery |
| topic | Optimization and Control 47A52 G.1.6 |
| url | https://arxiv.org/abs/2506.11372 |