The accelerated tensor Kaczmarz algorithm with adaptive parameters for solving tensor systems

Solving tensor systems is a common task in scientific computing and artificial intelligence. In this paper, we propose a tensor randomized average Kaczmarz method with adaptive parameters that exponentially converges to the unique least Frobenius norm solution of a given consistent tensor system under the t-product structure. In order to accelerate convergence, a tensor average Kaczmarz method based on stochastic heavy ball momentum technique (tAKSHBM) is proposed. The tAKSHBM method utilizes iterative information to update parameters instead of relying on prior information, addressing the problem in the adaptive learning of parameters. Additionally, the tAKSHBM method based on Fourier transform is proposed, which can be effectively implemented in a distributed environment. It is proven that the iteration sequences generated by all the proposed methods are convergent for given consistent tensor systems. Finally, we conduct experiments on both synthetic data and practical applications to support our theoretical results and demonstrate the effectiveness of the proposed algorithms.