  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 provide convergence guarantees for the
proposed tensor randomized Kaczmarz that are analogous to those of the
randomized Kaczmarz method for matrix linear systems. 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.
