Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910586476953600 |
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| author | Sweeney, Brian Español, Malena I. Renaut, Rosemary |
| author_facet | Sweeney, Brian Español, Malena I. Renaut, Rosemary |
| contents | We consider an $\ell_1$-regularized inverse problem where both the forward and regularization operators have a Kronecker product structure. By leveraging this structure, a joint decomposition can be obtained using generalized singular value decompositions. We show how this joint decomposition can be effectively integrated into the Split Bregman and Majorization-Minimization methods to solve the $\ell_1$-regularized inverse problem. Furthermore, for cases involving column-orthogonal regularization matrices, we prove that the joint decomposition can be derived directly from the singular value decomposition of the system matrix. As a result, we show that framelet and wavelet operators are efficient for these decomposition-based algorithms in the context of $\ell_1$-regularized image deblurring problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00883 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices Sweeney, Brian Español, Malena I. Renaut, Rosemary Numerical Analysis 65F22, 65F10, 68W40 We consider an $\ell_1$-regularized inverse problem where both the forward and regularization operators have a Kronecker product structure. By leveraging this structure, a joint decomposition can be obtained using generalized singular value decompositions. We show how this joint decomposition can be effectively integrated into the Split Bregman and Majorization-Minimization methods to solve the $\ell_1$-regularized inverse problem. Furthermore, for cases involving column-orthogonal regularization matrices, we prove that the joint decomposition can be derived directly from the singular value decomposition of the system matrix. As a result, we show that framelet and wavelet operators are efficient for these decomposition-based algorithms in the context of $\ell_1$-regularized image deblurring problems. |
| title | Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices |
| topic | Numerical Analysis 65F22, 65F10, 68W40 |
| url | https://arxiv.org/abs/2409.00883 |