Deterministic and randomized Kaczmarz methods for $AXB=C$ with applications to color image restoration

TL;DR

This paper introduces new deterministic and randomized Kaczmarz algorithms for solving linear matrix equations, with proven convergence and applications to color image restoration, advancing iterative methods in matrix computations.

Contribution

The paper develops and analyzes novel block Kaczmarz methods, including greedy randomized and deterministic variants, for efficient solution of matrix equations with theoretical convergence guarantees.

Findings

Convergence of proposed methods established

Numerical tests confirm theoretical results

Effective application to color image restoration

Abstract

We study Kaczmarz type methods to solve consistent linear matrix equations. We first present a block Kaczmarz (BK) method that employs a deterministic cyclic row selection strategy. Assuming that the associated coefficient matrix has full column or row rank, we derive matrix formulas for a cycle of this BK method. Moreover, we propose a greedy randomized block Kaczmarz (GRBK) method and further extend it to a relaxed variant (RGRBK) and a deterministic counterpart (MWRBK). We establish the convergence properties of the proposed methods. Numerical tests verify the theoretical findings, and we apply the proposed methods to color image restoration problems.