Randomized iterative algorithms, such as the randomized Kaczmarz method, have gained considerable popularity due to their efficacy in solving matrix-vector and matrix-matrix regression problems. Our present work leverages the insights gained from studying such algorithms to develop regression methods for tensors, which are the natural setting for many application problems, e.g., image deblurring. In particular, we extend the randomized Kaczmarz method to solve a tensor system of the form $\mathbf{\mathcal{A}}\mathcal{X} = \mathcal{B}$, where $\mathcal{X}$ can be factorized as $\mathcal{X} = \mathcal{U}\mathcal{V}$, and all products are calculated using the t-product. We develop variants of the randomized factorized Kaczmarz method for matrices that approximately solve tensor systems in both the consistent and inconsistent regimes. We provide theoretical guarantees of the exponential convergence rate of our algorithms, accompanied by illustrative numerical simulations. Furthermore, we situate our method within a broader context by linking our novel approaches to earlier randomized Kaczmarz methods.

翻译：随机化迭代算法，如随机化Kaczmarz方法，因其在求解矩阵-向量和矩阵-矩阵回归问题中的高效性而广受欢迎。本研究借鉴了对此类算法的研究成果，开发了面向张量的回归方法，张量是许多应用问题（如图像去模糊）的自然建模框架。具体而言，我们将随机化Kaczmarz方法推广至求解形式为$\mathbf{\mathcal{A}}\mathcal{X} = \mathcal{B}$的张量系统，其中$\mathcal{X}$可因子化为$\mathcal{X} = \mathcal{U}\mathcal{V}$，且所有乘积均使用t-积进行计算。我们针对矩阵开发了随机化因子化Kaczmarz方法的多种变体，用于在相容与不相容情形下近似求解张量系统。我们为算法的指数收敛速率提供了理论保证，并辅以说明性的数值模拟。此外，通过将我们的新方法与早期的随机化Kaczmarz方法相联系，我们将本方法置于更广泛的背景中进行讨论。