When solving noisy linear systems Ax = b + c, the theoretical and empirical performance of stochastic iterative methods, such as the Randomized Kaczmarz algorithm, depends on the noise level. However, if there are a small number of highly corrupt measurements, one can instead use quantile-based methods to guarantee convergence to the solution x of the system, despite the presence of noise. Such methods require the computation of the entire residual vector, which may not be desirable or even feasible in some cases. In this work, we analyze the sub-sampled quantile Randomized Kaczmarz (sQRK) algorithm for solving large-scale linear systems which utilize a sub-sampled residual to approximate the quantile threshold. We prove that this method converges to the unique solution to the linear system and provide numerical experiments that support our theoretical findings. We additionally remark on the extremely small sample size case and demonstrate the importance of interplay between the choice of quantile and subset size.

翻译：在求解含噪线性系统Ax = b + c时，随机迭代方法（如随机Kaczmarz算法）的理论与实证性能依赖于噪声水平。然而，当存在少量高度污染的测量值时，可采用基于分位数的方法确保系统解x的收敛性，即使存在噪声干扰。这类方法需要计算完整残差向量，这在某些情况下可能不可取甚至不可行。本文分析了用于求解大规模线性系统的子采样分位数随机Kaczmarz（sQRK）算法，该算法利用子采样残差近似分位数阈值。我们证明该方法收敛至线性系统的唯一解，并通过数值实验支持理论结果。此外，我们针对极小样本量情形进行了讨论，展示了分位数选择与子集大小之间相互作用的的重要性。