The Kaczmarz method is a popular iterative scheme for solving large-scale linear systems. The randomized Kaczmarz method (RK) greatly improves the convergence rate of the Kaczmarz method, by using the rows of the coefficient matrix in random order rather than in their given order. An obvious disadvantage of the randomized Kaczmarz method is its probability criterion for selecting the active or working rows in the coefficient matrix. In [{\sc Z.Z. Bai, W. Wu}, {\em On greedy randomized Kaczmarz method for solving large sparse linear systems}, SIAM Journal on Scientific Computing, 2018, 40: A592--A606], the authors proposed a greedy randomized Kaczmarz method (GRK). However, this method may suffer from heavily computational cost when the size of the matrix is large, and the overhead will be prohibitively large for big data problems. The contribution of this work is as follows. First, from the probability significance point of view, we present a partially randomized Kaczmarz method, which can reduce the computational overhead needed in greedy randomized Kaczmarz method. Second, based on Chebyshev's law of large numbers and Z-test, we apply a simple sampling approach to the partially randomized Kaczmarz method. The convergence of the proposed method is established. Third, we apply the new strategy to the ridge regression problem, and propose a partially randomized Kaczmarz method with simple random sampling for ridge regression. Numerical experiments demonstrate the superiority of the new algorithms over many state-of-the-art randomized Kaczmarz methods for large linear systems problems and ridge regression problems.

翻译：Kaczmarz 法是解决大规模线性系统的流行的迭代机制。 随机的 Kaczmarz 法( RK ) 通过随机使用系数矩阵的行, 而不是按给定的顺序, 大大提高了Kaczmarz 法的趋同率率。 随机的Kaczmarz 法的一个明显缺点是其在系数矩阵中选择活动行或工作行的概率标准。 在 [ SSc Z. Bai, W. W. Wu} ) 中, 贪婪的 随机的卡兹马尔兹 法, 用于解决大规模分散的线性系统 }, SIAM 科学计算学期刊, 2018, 40: A592- A606, 作者们提出了一种贪婪的随机卡兹马兹 矩阵矩阵方法( GRGRK ) 。 然而, 这种方法可能因在矩阵大小时的计算成本巨大而受到影响, 而高得令人望的大数据问题。 这项工作的贡献如下。 首先, 我们从概率意义的角度展示了部分的 Kaczmarz 方法, 。, 将部分的卡兹 方法, 将 Kac 的计算法 方法 的计算法方法 用于我们 的 的 的 的 的 的 的大规模 法 方法 。