发表机构
Virginia Tech; Tufts University（弗吉尼亚理工大学; 塔夫茨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对大规模反问题，提出一种可证明收敛的MM-GKS变体，通过交替压缩和扩展搜索空间，在保持小维度下严格单调收敛，显著降低内存与计算成本，并推广至流式数据处理。
AI 中文摘要
对于具有清晰边缘的高质量图像，边缘保持正则化的一种流行选择是使用图像梯度的广义$\ell_q$范数。这可以通过$\ell_2$范数和一系列加权梯度来高效实现，其中权重由当前解估计导出。我们可以使用混合Krylov子空间方法求解由此产生的正则化最小二乘问题序列，该方法利用投影到Krylov子空间上的问题高效计算正则化参数。然而，正则化算子的每次更新都需要一个新的Krylov子空间。最小化-最大化广义Krylov子空间方法（MM-GKS）通过使用单一的广义Krylov子空间（GKS）解决了这一问题。不幸的是，对于大规模问题，如果收敛速度不快，MM-GKS会带来巨大的内存需求和计算成本。我们提出了一种MM-GKS的变体，它在保持严格单调收敛的同时交替压缩和扩展搜索空间。我们证明，即使搜索空间维度保持很小，我们的方法也能可证明地收敛到所选泛函的最小值。这大大改进了MM-GKS先前的理论结果，后者的收敛性证明依赖于解空间基（最终）张成整个空间。我们表明，我们的方法在内存需求和计算复杂度方面都能高效解决大规模问题。我们进一步将所提出的方法推广到处理流式问题，即数据要么不能同时全部获得，要么由于极端的内存需求而需要如此处理。我们使用来自图像去模糊、动态光声断层扫描和流式X射线计算机断层扫描（CT）的数值示例来说明我们提出方法的有效性。
英文摘要
For high-quality images with sharp edges, a popular choice for edge-preserving regularization is using a general(ized) $\ell_q$-norm of the gradient of the image. This can be implemented efficiently using the $\ell_2$-norm and a sequence of weighted gradients, with weights derived from the current solution estimate. We can solve the resulting sequence of regularized least squares problems using hybrid Krylov subspace methods, which efficiently compute the regularization parameter using the problem projected on the Krylov subspace. However, each update of the regularization operator requires a new Krylov subspace. The majorization-minimization generalized Krylov subspace method (MM-GKS) addresses this problem by using a single, generalized, Krylov subspace (GKS). Unfortunately, for large-scale problems, if convergence is not fast, MM-GKS has overwhelming memory requirements and computational costs. We propose a variant of MM-GKS that alternately compresses and expands the search space while maintaining strict monotonic convergence. We show that our method provably converges to the minimum of the selected functional, even if the search space dimension is kept very small. This substantially improves on previous theoretical results for MM-GKS, where the convergence proof relies on the basis for the solution space (eventually) spanning the full space. We show that our method can solve large-scale problems efficiently both in terms of memory requirements and computational complexity. We further generalize our proposed method to handle streaming problems, where the data is either not all available simultaneously or needs to be treated as such because of the extreme memory requirements. We use numerical examples from image deblurring, dynamic photoacoustic tomography, and streaming X-ray computed tomography (CT) to illustrate the effectiveness of our proposed methods.
Comments26 pages, 9 figures, 4 tables
