Sparsity-Guided Multi-Parameter Selection in $\ell_1$-Regularized Models via a Fixed-Point Proximity Approach
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908799297650688 |
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| author | Liu, Qianru Wang, Rui Xu, Yuesheng |
| author_facet | Liu, Qianru Wang, Rui Xu, Yuesheng |
| contents | We study a regularization framework that combines a convex fidelity term with multiple $\ell_1$-based regularizers, each linked to a distinct linear transform. This multi-penalty model enhances flexibility in promoting structured sparsity. We analyze how the choice of regularization parameters governs the sparsity of solutions under the given transforms and derive a precise relationship between the parameters and resulting sparsity patterns. This insight enables the development of an iterative strategy for selecting parameters to achieve prescribed sparsity levels. A key computational challenge arises in practice: effective parameter tuning requires simultaneous access to the regularized solution and two auxiliary vectors derived from the sparsity analysis. To address this, we propose a fixed-point proximity algorithm that jointly computes all three vectors. Together with our theoretical characterization, this algorithm forms the basis of a practical multi-parameter selection scheme. Numerical experiments demonstrate that the proposed method reliably produces solutions with desired sparsity patterns and strong approximation accuracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00655 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sparsity-Guided Multi-Parameter Selection in $\ell_1$-Regularized Models via a Fixed-Point Proximity Approach Liu, Qianru Wang, Rui Xu, Yuesheng Numerical Analysis We study a regularization framework that combines a convex fidelity term with multiple $\ell_1$-based regularizers, each linked to a distinct linear transform. This multi-penalty model enhances flexibility in promoting structured sparsity. We analyze how the choice of regularization parameters governs the sparsity of solutions under the given transforms and derive a precise relationship between the parameters and resulting sparsity patterns. This insight enables the development of an iterative strategy for selecting parameters to achieve prescribed sparsity levels. A key computational challenge arises in practice: effective parameter tuning requires simultaneous access to the regularized solution and two auxiliary vectors derived from the sparsity analysis. To address this, we propose a fixed-point proximity algorithm that jointly computes all three vectors. Together with our theoretical characterization, this algorithm forms the basis of a practical multi-parameter selection scheme. Numerical experiments demonstrate that the proposed method reliably produces solutions with desired sparsity patterns and strong approximation accuracy. |
| title | Sparsity-Guided Multi-Parameter Selection in $\ell_1$-Regularized Models via a Fixed-Point Proximity Approach |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2502.00655 |