The Randomized Kaczmarz method (RK) is a stochastic iterative method for solving linear systems that has recently grown in popularity due to its speed and low memory requirement. Selectable Set Randomized Kaczmarz (SSRK) is an variant of RK that leverages existing information about the Kaczmarz iterate to identify an adaptive ``selectable set'' and thus yields an improved convergence guarantee. In this paper, we propose a general perspective for selectable set approaches and prove a convergence result for that framework. In addition, we define two specific selectable set sampling strategies that have competitive convergence guarantees to those of other variants of RK. One selectable set sampling strategy leverages information about the previous iterate, while the other leverages the orthogonality structure of the problem via the Gramian matrix. We complement our theoretical results with numerical experiments that compare our proposed rules with those existing in the literature.

翻译：随机的 Kaczmarz 方法( RK) 是解决线性系统的一种随机迭代方法, 最近由于速度和内存要求低而越来越受欢迎。 选择的 Set 随机化 Kaczmarz (SSRK) 是RK 的变体, 利用关于 Kaczmarz 循环的现有信息来确定适应性“ 可选择的一组”, 从而产生更好的趋同保证。 在本文件中, 我们为可选择的一套办法提出了一个总体观点, 并证明这个框架的趋同结果。 此外, 我们定义了两种具体的可选择的成套抽样战略, 与RK的其他变体具有竞争性的趋同保证。 一种可选择的抽样战略利用关于前一个变体的信息, 而其他的抽样战略则利用格拉姆矩阵来利用问题的正方形结构。 我们的理论结果是用数字实验来补充我们的理论结果, 将我们提议的规则与文献中的现有规则进行比较。