On the randomized block Kaczmarz algorithms for solving matrix equation AXB=C

Abstract

The randomized Kaczmarz algorithm is one of the most popular approaches for solving large-scale linear systems due to its simplicity and efficiency. In this paper, we introduce two classes of randomized Kaczmarz methods for solving linear matrix equations $AXB=C$: the randomized block Kaczmarz (ME-RBK) and randomized average block Kaczmarz (ME-RABK) algorithms. The key feature of the ME-RBK algorithm is that the current iterate is projected onto the solution space of the sketched matrix equation at each iteration. In contrast, the ME-RABK method is pseudoinverse-free, enabling deployment on parallel computing units to significantly reduce computational time. We demonstrate that these two methods converge linearly in the mean square to the minimum norm solution $X_{\ast }=A^{†}C{B}^{†}$ of a given linear matrix equation. The convergence rates are influenced by the geometric properties of the data matrices and their submatrices, as well as by the block sizes. Numerical results indicate that our proposed algorithms are both efficient and effective for solving linear matrix equations, particularly excelling in image deblurring applications.