On greedy randomized Kaczmarz-type methods for solving the system of tensor equations

Abstract

For solving the system of tensor equations $A{x}^{m-1}=b$, where $\mathrm{x,b}\in R^n$ and $A$ is an $m$-order $n$-dimensional real tensor, we introduce two greedy Kaczmarz-type methods: the tensor relaxed greedy randomized Kaczmarz algorithm and the accelerated tensor relaxed greedy Kaczmarz algorithm. The deterministic convergence analysis of both methods is given based on the local tangential cone condition. Numerical results demonstrate that the greedy Kaczmarz-type methods are more efficient than the randomized Kaczmarz-type methods, and the accelerated greedy version exhibits significant acceleration.