Randomized block coordinate descent type methods have been demonstrated to be efficient for solving large-scale optimization problems. Linear convergence to the unique solution is established if the objective function is strongly convex. In this paper we propose a randomized block coordinate descent algorithm for solving the matrix least squares problem ${min}_{\mathbf{X}\in {\mathbb{R}}^{m\times n}}{\Vert \mathbf{C\; -\; AXB}\Vert }_{\mathrm{F}}$ with $\mathbf{A}\in {\mathbb{R}}^{p\times m}$, $\mathbf{B}\in {\mathbb{R}}^{n\times q}$, and $\mathbf{C}\in {\mathbb{R}}^{p\times q}$. We prove that the proposed algorithm converges linearly to the unique minimum norm least squares solution (i.e., ${\mathbf{A}}^{†}{\mathbf{C}\mathbf{B}}^{†}$) without the strong convexity assumption. Instead, we only need that $\mathbf{B}$ has full row rank. Numerical experiments are given to illustrate the theoretical results.

随机块坐标下降类型方法已被证明对于解决大规模优化问题是有效的。如果目标函数是强凸的，则建立到唯一解的线性收敛。在本文中，我们提出了一种用于解决矩阵最小二乘问题的随机块坐标下降算法${分钟}_{\mathbf{X}\in {\mathbb{电阻}}^{米\times n}}{\Vert \mathbf{C-AXB}\Vert }_{\mathrm{F}}$ 和 $\mathbf{一种}\in {\mathbb{电阻}}^{磷\times 米}$, $\mathbf{乙}\in {\mathbb{电阻}}^{n\times q}$， 和 $\mathbf{C}\in {\mathbb{电阻}}^{磷\times q}$. 我们证明了所提出的算法线性收敛到唯一的最小范数最小二乘解（即${\mathbf{一种}}^{†}{\mathbf{C}\mathbf{乙}}^{†}$) 没有强凸性假设。相反，我们只需要$\mathbf{乙}$具有完整的行等级。给出了数值实验来说明理论结果。