Quantile Randomized Kaczmarz Algorithm with Whitelist Trust Mechanism

TL;DR

This paper improves the robustness and efficiency of the QuantileRK algorithm for solving noisy overdetermined linear systems by introducing an online detection mechanism and quantile estimation from small samples.

Contribution

It reanalyzes QRK's convergence, proposes a practical online detector for unreliable rows, and reduces computational cost via subsampling.

Findings

Convergence rate improves as corruption decreases.

The online detector effectively flags unreliable equations.

Subsampling preserves robustness while lowering per-iteration cost.

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 $\|\boldsymbol{\varepsilon}\|_0 \le \beta m$. 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 $\beta$ decreases; (ii) propose a simple online detector that flags and removes unreliable rows, which reduces the effective $\beta$ 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.