Quantile-based Random Sparse Kaczmarz for Corrupted and Noisy Linear Systems

TL;DR

This paper introduces a quantile-based randomized sparse Kaczmarz method with linear convergence guarantees, designed to effectively handle corrupted and noisy large-scale linear systems, and enhances efficiency through parallelization.

Contribution

It develops a novel quantile-based variant of the randomized sparse Kaczmarz method with proven convergence and incorporates block techniques for parallel acceleration.

Findings

The method achieves linear convergence even with heavy corruptions.

Numerical experiments demonstrate high efficiency and robustness.

Parallel implementation significantly speeds up computations.

Abstract

The randomzied Kaczmarz method, along with its recently developed variants, has become a popular tool for dealing with large-scale linear systems. However, these methods usually fail to converge when the linear systems are affected by heavy corruptions, which are common in many practical applications. In this study, we develop a new variant of the randomzied sparse Kaczmarz method with linear convergence guarantees, by making use of a quantile technique to detect corruptions. Moreover, we incorporate averaged block technique into the proposed method to achieve parallel computation and acceleration. Finally, the proposed algorithms are illustrated to be very efficient through extensive numerical experiments.