标题： 带有分解算子的t-乘积张量线性系统的随机Kaczmarz方法 显示英文标题

标题： Randomized Kaczmarz methods for t-product tensor linear systems with factorized operators

摘要： 随机迭代算法，如随机Kaczmarz方法，由于在解决矩阵-向量和矩阵-矩阵回归问题方面的有效性而获得了相当大的流行度。 我们目前的工作利用了研究此类算法所得的见解，以开发适用于张量的回归方法，这些张量是许多应用问题的自然设置，例如图像去模糊。 具体来说，我们将随机Kaczmarz方法扩展为求解形式为$\mathbf{\mathcal{A}}\mathcal{X} = \mathcal{B}$的张量系统，其中$\mathcal{X}$可以分解为$\mathcal{X} = \mathcal{U}\mathcal{V}$，且所有乘积均使用t-乘积计算。 我们开发了针对矩阵的随机分解Kaczmarz方法的变体，在一致和不一致情况下近似求解张量系统。 我们提供了算法指数收敛速率的理论保证，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Kaczmarz方法联系起来，我们将我们的方法置于一个更广泛的背景下。 显示英文摘要

摘要： 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.