On global randomized block Kaczmarz algorithm for solving large-scale matrix equations

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 propose two classes of global randomized Kaczmarz methods for solving large-scale linear matrix equations $AXB=C$, the global randomized block Kaczmarz algorithm and global randomized average block Kaczmarz algorithm. The feature of global randomized block Kaczmarz algorithm is the fact that the current iterate is projected onto the solution space of the sketched matrix equation at each iteration, while the global randomized average block Kaczmarz approach is pseudoinverse-free and therefore can be deployed on parallel computing units to achieve significant improvements in the computational time. We prove that these two methods linearly converge in the mean square to the minimum norm solution $X_*=A^\dag CB^\dag$ of a given linear matrix equation. The convergence rates depend on the geometric properties of the data matrices and their submatrices and on the size of the blocks. Numerical results reveal that our proposed algorithms are efficient and effective for solving large-scale matrix equations. In particular, they can also achieve satisfying performance when applied to image deblurring problems.