Quantile-based Iterative Methods for Corrupted Systems of Linear Equations

TL;DR

This paper introduces quantile-based iterative algorithms designed to accurately solve large-scale linear systems despite significant measurement corruptions, with proven convergence and empirical validation.

Contribution

It proposes novel iterative methods utilizing residual quantiles to robustly recover solutions from corrupted linear systems, advancing beyond traditional techniques.

Findings

Methods converge to uncorrupted solutions despite corruptions

Theoretical guarantees of convergence are established

Empirical results validate robustness and effectiveness

Abstract

Often in applications ranging from medical imaging and sensor networks to error correction and data science (and beyond), one needs to solve large-scale linear systems in which a fraction of the measurements have been corrupted. We consider solving such large-scale systems of linear equations $\mathbf{A}\mathbf{x}=\mathbf{b}$ that are inconsistent due to corruptions in the measurement vector $\mathbf{b}$. We develop several variants of iterative methods that converge to the solution of the uncorrupted system of equations, even in the presence of large corruptions. These methods make use of a quantile of the absolute values of the residual vector in determining the iterate update. We present both theoretical and empirical results that demonstrate the promise of these iterative approaches.