Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices
<|reference_start|>Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices: 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.<|reference_end|>
@article{sweeney2024efficient,
    title={Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized
  Inverse Problems with Column-Orthogonal and Kronecker Product Matrices},
    author={Brian Sweeney, Malena I. Espa~nol, Rosemary Renaut},
    journal={arXiv preprint arXiv:2409.00883},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.00883},
    primaryClass={math.NA cs.NA}
}