基于列正交与Kronecker积矩阵的ℓ1正则化反问题的高效分解算法
Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices
AI总结：
本文针对具有Kronecker积结构的ℓ1正则化反问题，提出利用广义奇异值分解构造联合分解，并将其集成到Split Bregman和Majorization-Minimization方法中，同时证明列正交情形下可直接由系统矩阵奇异值分解导出联合分解，从而高效处理图像去模糊问题。
摘要：
我们考虑一个ℓ1正则化反问题，其中前向算子和正则化算子都具有Kronecker积结构。通过利用这一结构，可以使用广义奇异值分解获得联合分解。我们展示了如何将该联合分解有效地集成到Split Bregman方法和Majorization-Minimization方法中，以求解该ℓ1正则化反问题。此外，对于涉及列正交正则化矩阵的情形，我们证明联合分解可以直接从系统矩阵的奇异值分解导出。因此，我们表明，在ℓ1正则化图像去模糊问题的背景下，framelet和小波算子对于这些基于分解的算法是高效的。
英文摘要：
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.
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