Relaxed Greedy Randomized Kaczmarz with Signal Averaging for Solving Doubly-Noisy Linear Systems

TL;DR

This paper introduces RGRK-SA, an enhanced algorithm for solving large, doubly-noisy linear systems, which converges faster and more accurately by averaging multiple measurements.

Contribution

It extends the RGRK method to doubly-noisy systems and proposes a simple averaging modification that guarantees convergence to the least-squares solution.

Findings

RGRK-SA converges at a polynomial rate for doubly-noisy systems.

RGRK-SA outperforms classical methods in accuracy.

Numerical experiments confirm improved convergence and accuracy.

Abstract

Large-scale linear systems of the form $Ax=b$ are often doubly-noisy, in the sense that both its measurement matrix $A$ and measurement vector $b$ are noisy. In this paper, we extend the relaxed greedy randomized Kaczmarz (RGRK) method to the doubly-noisy systems to accelerate convergence. However, RGRK fails to converge to the least-squares solution for doubly-noisy systems. To address this limitation, we propose a simple modification: averaging multiple measurements instead of using a single measurement. The proposed RGRK with signal averaging (RGRK-SA) converges to the solution of doubly-noisy systems at a polynomial rate. Numerical experiments demonstrate that both RGRK and RGRK-SA outperform the classical randomized Kaczmarz method, and RGRK-SA has a higher accuracy.