Reverse Quantile-RK and its Application to Quantile-RK

TL;DR

This paper introduces a new accelerated randomized Kaczmarz method leveraging quantile information to efficiently and robustly solve large, corrupted linear systems, outperforming existing algorithms in convergence speed.

Contribution

It proposes a novel acceleration technique for the randomized Kaczmarz method using quantile data, enhancing convergence and robustness without extra computational cost.

Findings

Faster convergence than standard randomized Kaczmarz

Robustness to large, sparse corruptions in measurements

Effective in large-scale, corrupted linear systems

Abstract

When solving linear systems $Ax=b$, $A$ and $b$ are given, but the measurements $b$ often contain corruptions. Inspired by recent work on the quantile-randomized Kaczmarz method, we propose an acceleration of the randomized Kaczmarz method using quantile information. We show that the proposed acceleration converges faster than the randomized Kaczmarz algorithm. In addition, we show that our proposed approach can be used in conjunction with the quantile-randomized Kaczamrz algorithm, without adding additional computational complexity, to produce both a fast and robust iterative method for solving large, sparsely corrupted linear systems. Our extensive experimental results support the use of the revised algorithm.