For solving the large-scale linear least-squares problem, we propose a block version of the randomized extended Kaczmarz method, called the two-subspace randomized extended Kaczmarz method, which does not require any row or column paving. Theoretical analysis and numerical results show that the two-subspace randomized extended Kaczmarz method is much more efficient than the randomized extended Kaczmarz method. When the coefficient matrix is of full column rank, the two-subspace randomized extended Kaczmarz method can also outperform the randomized coordinate descent method. If the linear system is consistent, we remove one of the iteration sequences in the two-subspace randomized extended Kaczmarz method, which approximates the projection of the right-hand side vector onto the orthogonal complement space of the range space of the coefficient matrix, and obtain the generalized two-subspace randomized Kaczmarz method, which is actually a generalization of the two-subspace randomized Kaczmarz method without the assumptions of unit row norms and full column rank on the coefficient matrix. We give the upper bound for the convergence rate of the generalized two-subspace randomized Kaczmarz method which also leads to a better upper bound for the convergence rate of the two-subspace randomized Kaczmarz method.

为了解决大规模线性最小二乘问题，我们提出了一种随机扩展的扩展Kaczmarz方法的块版本，称为两子空间随机扩展的Kaczmarz方法，它不需要任何行或列铺装。理论分析和数值结果表明，两子空间随机扩展Kaczmarz方法比随机扩展Kaczmarz方法更有效。当系数矩阵为全列秩时，两子空间随机扩展Kaczmarz方法也可以胜过随机坐标下降法。如果线性系统是一致的，我们将在两子空间随机扩展的Kaczmarz方法中删除迭代序列之一，近似将右侧向量投影到系数矩阵范围空间的正交补空间上，并得到广义的两个子空间随机化的Kaczmarz方法，实际上是对两个子空间的随机化Kaczmarz方法的一个泛化。系数矩阵上的单位行范数和完整列秩的假设。我们给出了广义两子空间随机Kaczmarz方法的收敛速度的上限，这也导致了两子空间随机Kaczmarz方法的收敛速度的更好的上限。