Solving linear systems of equations is a fundamental problem in mathematics. When the linear system is so large that it cannot be loaded into memory at once, iterative methods such as the randomized Kaczmarz method excel. Here, we extend the randomized Kaczmarz method to solve multi-linear (tensor) systems under the tensor–tensor t-product. We present convergence guarantees for tensor randomized Kaczmarz in two ways: using the classical matrix randomized Kaczmarz analysis and taking advantage of the tensor–tensor t-product structure. We demonstrate experimentally that the tensor randomized Kaczmarz method converges faster than traditional randomized Kaczmarz applied to a naively matricized version of the linear system. In addition, we draw connections between the proposed algorithm and a previously known extension of the randomized Kaczmarz algorithm for matrix linear systems.

求解线性方程组是数学中的一个基本问题。当线性系统太大而无法立即加载到内存中时，诸如随机化的Kaczmarz方法之类的迭代方法非常出色。在这里，我们扩展了随机的Kaczmarz方法，以求解张量-张量t积下的多线性（张量）系统。我们以两种方式为张量随机Kaczmarz提供收敛保证：使用经典矩阵随机Kaczmarz分析并利用张量-张量t积结构。我们通过实验证明，张量随机Kaczmarz方法的收敛速度快于应用于线性系统的朴素矩阵版本的传统随机Kaczmarz。此外，