We study a subspace constrained version of the randomized Kaczmarz algorithm for solving large linear systems in which the iterates are confined to the space of solutions of a selected subsystem. We show that the subspace constraint leads to an accelerated convergence rate, especially when the system has structure such as having coherent rows or being approximately low-rank. On Gaussian-like random data, it results in a form of dimension reduction that effectively improves the aspect ratio of the system. Furthermore, this method serves as a building block for a second, quantile-based algorithm for the problem of solving linear systems with arbitrary sparse corruptions, which is able to efficiently exploit partial external knowledge about uncorrupted equations and achieve convergence in difficult settings such as in almost-square systems. Numerical experiments on synthetic and real-world data support our theoretical results and demonstrate the validity of the proposed methods for even more general data models than guaranteed by the theory.

翻译：我们研究了随机Kaczmarz算法的一种子空间约束变体，用于求解大型线性系统，其中迭代被限制在所选子系统的解空间内。我们证明，子空间约束能够加速收敛速度，尤其当系统具有结构特性（如行相干或近似低秩）时。在类高斯随机数据上，该方法实现了一种维度缩减，有效改善了系统的宽高比。此外，该方法可作为第二个基于分位数的算法的构建模块，用于求解带有任意稀疏污染物的线性系统问题，该算法能够高效利用关于未污染方程的部分外部知识，并在困难场景（如近方阵系统）中实现收敛。在合成数据和真实数据上的数值实验支持了我们的理论结果，并验证了所提方法在比理论保证更广泛的数据模型上的有效性。