Large-scale systems of linear equations arise in machine learning, medical imaging, sensor networks, and in many areas of data science. When the scale of the systems are extreme, it is common for a fraction of the data or measurements to be corrupted. The Quantile Randomized Kaczmarz (QRK) method is known to converge on large-scale systems of linear equations Ax=b that are inconsistent due to static corruptions in the measurement vector b. We prove that QRK converges even for systems corrupted by time-varying perturbations. Additionally, we prove that QRK converges up to a convergence horizon on systems affected by time-varying noise and demonstrate that the noise affects only the convergence horizon of the method, and not the rate of convergence. Finally, we utilize Markov's inequality to prove a lower bound on the probability that the largest entries of the QRK residual reveal the time-varying corruption in each iteration. We present numerical experiments which illustrate our theoretical results.

翻译：大规模线性方程组出现在机器学习、医学成像、传感器网络以及数据科学的众多领域。当系统规模极大时，部分数据或测量值常会受到扰动。已知分位数随机卡茨马兹（QRK）方法可收敛于因测量向量b存在静态扰动而不相容的大规模线性方程组Ax=b。我们证明QRK方法甚至在受时变扰动影响的系统中也能收敛。此外，我们证明QRK方法在受时变噪声影响的系统中能收敛至一个收敛界，并表明噪声仅影响该方法的收敛界，而不影响收敛速率。最后，我们利用马尔可夫不等式证明了下界，表明QRK残差的最大条目在每个迭代中以较高概率揭示出时变扰动。我们通过数值实验验证了理论结果。