On the linear convergence of circumcentered isometry methods
On the linear convergence of circumcentered isometry methods
复制标题
外心等距法的线性收敛性
DOI：
10.1007/s11075-020-00966-x
复制
发表时间：
2019
影响因子：
2.1
通讯作者：
Xianfu Wang
中科院分区：
文献类型：
--
作者：
Heinz H. Bauschke;Hui Ouyang;Xianfu Wang
由 Behling、Bello Cruz 和 Santos 提出的外心道格拉斯-拉赫福德方法 (C-DRM) 通过取相关连续反射的外心进行迭代。它是著名的 Douglas-Rachford 方法 (DRM) 的加速版本，用于寻找有限多个仿射子空间交集的最佳近似。受C-DRM的启发，我们引入了更灵活的外心反射法（CRM）和外心等距法（CIM）。 CIM 本质上是在关联仿射壳中的所有点中选择最接近解的点作为其迭代，并且是 CRM 的推广。 Behling、Bello Cruz 和 Santos 提出的用于推广 C-DRM 的外心反射方法是我们 CRM 的一个特殊类别。我们考虑由一组有限多个等距引起的 CIM，以找到等距定点集交集的最佳近似，该交集结果是有限多个仿射子空间的交集。我们将之前有限维空间中 CRM 的线性收敛结果从反射扩展到等距。为了更好地加速交替投影对称法（MAP），加速对称MAP首先对初始点应用另一个算子。 （类似地，为了加速 DRM，C-DRM 首先也将另一个算子应用于初始点。）受这些事实的启发，我们展示了希尔伯特空间中 CIM 线性收敛的结果，首先将另一个算子应用于初始点。特别是，在某些限制下，我们的结果表明某些 CRM 达到了希尔伯特空间中加速对称 MAP 的已知线性收敛速度。我们还展示了一类在希尔伯特空间中收敛到最佳近似的 CRM，其收敛速度不比 MAP 的急剧收敛速度差。一些 CRM 达到 MAP 线性收敛速度或加速对称 MAP 的事实是全新的。
The circumcentered Douglas–Rachford method (C–DRM), introduced by Behling, Bello Cruz and Santos, iterates by taking the circumcenter of associated successive reflections. It is an acceleration of the well-known Douglas-Rachford method (DRM) for finding the best approximation onto the intersection of finitely many affine subspaces. Inspired by the C–DRM, we introduced the more flexible circumcentered reflection method (CRM) and circumcentered isometry method (CIM). The CIM essentially chooses the closest point to the solution among all of the points in an associated affine hull as its iterate and is a generalization of the CRM. The circumcentered–reflection method introduced by Behling, Bello Cruz and Santos to generalize the C–DRM is a special class of our CRM. We consider the CIM induced by a set of finitely many isometries for finding the best approximation onto the intersection of fixed point sets of the isometries which turns out to be an intersection of finitely many affine subspaces. We extend our previous linear convergence results on CRMs in finite-dimensional spaces from reflections to isometries. In order to better accelerate the symmetric method of alternating projections (MAP), the accelerated symmetric MAP first applies another operator to the initial point. (Similarly, to accelerate the DRM, the C–DRM first applies another operator to the initial point as well.) Motivated by these facts, we show results on the linear convergence of CIMs in Hilbert spaces with first applying another operator to the initial point. In particular, under some restrictions, our results imply that some CRMs attain the known linear convergence rate of the accelerated symmetric MAP in Hilbert spaces. We also exhibit a class of CRMs converging to the best approximation in Hilbert spaces with a convergence rate no worse than the sharp convergence rate of MAP. The fact that some CRMs attain the linear convergence rate of MAP or accelerated symmetric MAP is entirely new.
影响因子：
2.2
作者：
Behling, Roger;Bello-Cruz, J.-Yunier;Santos, Luiz-Rafael
通讯作者：
Santos, Luiz-Rafael
影响因子：
2.1
作者：
Behling, Roger;Bello-Cruz, Yunier;Santos, Luiz-Rafael
通讯作者：
Santos, Luiz-Rafael