Quantile Randomized Kaczmarz Algorithm with Whitelist Trust Mechanism

Abstract

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.