Randomized Kaczmarz (RK) is a simple and fast solver for consistent overdetermined systems, but it is known to be fragile under noise. We study overdetermined $m\times n$ linear systems with a sparse set of corrupted equations, $ {\bf A}{\bf x}^\star = {\bf b}, $where only $\tilde{\bf b} = {\bf b} + \boldsymbol{\varepsilon}$ is observed with $\|\boldsymbol{\varepsilon}\|_0 \le βm$. The recently introduced QuantileRK (QRK) algorithm addresses this issue by testing residuals against a quantile threshold, but computing a per-iteration quantile across many rows is costly. In this work we (i) reanalyze QRK and show that its convergence rate improves monotonically as the corruption fraction $β$ decreases; (ii) propose a simple online detector that flags and removes unreliable rows, which reduces the effective $β$ and speeds up convergence; and (iii) make the method practical by estimating quantiles from a small random subsample of rows, preserving robustness while lowering the per-iteration cost. Simulations on imaging and synthetic data demonstrate the efficiency of the proposed method.

翻译：随机Kaczmarz（RK）算法是求解相容超定方程组的一种简洁快速方法，但已知其在噪声条件下表现脆弱。我们研究具有稀疏被污染方程组的超定$m\times n$线性系统${\bf A}{\bf x}^\star = {\bf b}$，其中仅能观测到$\tilde{\bf b} = {\bf b} + \boldsymbol{\varepsilon}$且满足$\|\boldsymbol{\varepsilon}\|_0 \le βm$。近期提出的分位数随机Kaczmarz（QRK）算法通过将残差与分位数阈值进行比较来解决此问题，但跨多行计算每次迭代的分位数代价高昂。本工作中我们（i）重新分析QRK算法，证明其收敛速率随污染比例$β$减小而单调提升；（ii）提出一种简单的在线检测器，用于标记并移除不可靠行，从而降低有效$β$值并加速收敛；（iii）通过从随机小规模行子样本估计分位数使方法实用化，在保持鲁棒性的同时降低单次迭代成本。在成像与合成数据上的仿真实验验证了所提方法的效率。