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

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.

💡 Research Summary

**
The paper addresses the problem of solving consistent matrix equations of the form (AX B = C) by extending the classical Kaczmarz row‑action method to the matrix setting. The authors first introduce a deterministic Block Kaczmarz (BK) algorithm that cycles through the rows of (A) in a fixed order. At each iteration a rank‑one update is performed: \