On convergence of residual-based extended randomized Kaczmarz methods for matrix equations

TL;DR

This paper introduces residual-based randomized extended Kaczmarz methods for inconsistent matrix equations, providing convergence analysis without full column rank assumptions and demonstrating superior effectiveness through numerical experiments.

Contribution

It proposes new residual-based randomized Kaczmarz methods with convergence guarantees and momentum, extending applicability to inconsistent systems without full rank.

Findings

Methods outperform existing algorithms in numerical tests

Convergence bounds established without full column rank assumptions

Momentum enhances the effectiveness of the proposed methods

Abstract

In this paper, for solving inconsistent matrix equations we propose a dual-space residual-based randomized extended Kaczmarz method and its version with Nesterov momentum. Without the full column rank assumptions on coefficient matrices, we provide a thorough convergence analysis, and derive upper bounds for the convergence rates of the new methods. A feasible range for the momentum parameters is determined. Numerical experiments demonstrate that the proposed methods are much more effective than the existing ones, especially the method with momentum.