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On the linear convergence of circumcentered isometry methods
Numerical Algorithms ( IF 2.1 ) Pub Date : 2020-07-11 , DOI: 10.1007/s11075-020-00966-x
Heinz H. Bauschke , Hui Ouyang , Xianfu Wang

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.



中文翻译:

关于绕心等距方法的线性收敛

由Behling,Bello Cruz和Santos引入的以圆心为中心的Douglas–Rachford方法(C–DRM)通过采用相关的连续反射的圆心进行迭代。它是众所周知的Douglas-Rachford方法(DRM)的加速,用于在有限多个仿射子空间的交点上找到最佳逼近。在C–DRM的启发下,我们引入了更灵活的围绕中心的反射方法(CRM)和围绕中心的等轴测图方法(CIM)。CIM本质上是在关联仿射船体的所有点中选择与解决方案最接近的点作为其迭代项,并且是CRM的一种概括。Behling,Bello Cruz和Santos引入的围绕中心反射的方法来概括C-DRM是我们CRM的特殊类别。我们考虑由一组有限多个等距性诱导的CIM,以便找到等距的固定点集的交集的最佳近似值,结果证明是有限个仿射子空间的交集。我们将有限元空间中CRM的先前线性收敛结果扩展到从反射到对称。为了更好地加速交替投影的对称方法(MAP),加速的对称MAP首先将另一个算子应用于初始点。(同样,为了加速DRM,C–DRM首先也将另一个算子应用于初始点。)基于这些事实,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在Hilbert空间中获得加速对称MAP的已知线性收敛速度。我们还展示了收敛于希尔伯特空间中最佳近似的一类CRM,其收敛速度不比MAP的急剧收敛速度差。一些CRM达到MAP或加速对称MAP的线性收敛速度这一事实是全新的。

更新日期:2020-07-13
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