Randomized Average Kaczmarz Algorithm for Tensor Linear Systems

For solving tensor linear systems under the tensor–tensor t-product, we propose the randomized average Kaczmarz (TRAK) algorithm, the randomized average Kaczmarz algorithm with random sampling (TRAKS), and their Fourier version, which can be effectively implemented in a distributed environment. We analyzed the relationships (of the updated formulas) between the original algorithms and their Fourier versions in detail and prove that these new algorithms can converge to the unique least F-norm solution of the consistent tensor linear systems. Extensive numerical experiments show that they significantly outperform the tensor-randomized Kaczmarz (TRK) algorithm in terms of both iteration counts and computing times and have potential in real-world data, such as video data, CT data, etc.