Tensors serve as a crucial tool in the representation and analysis of complex, multi-dimensional data. As data volumes continue to expand, there is an increasing demand for developing optimization algorithms that can directly operate on tensors to deliver fast and effective computations. Many problems in real-world applications can be formulated as the task of recovering high-order tensors characterized by sparse and/or low-rank structures. In this work, we propose novel Kaczmarz algorithms with a power of the $\ell_1$-norm regularization for reconstructing high-order tensors by exploiting sparsity and/or low-rankness of tensor data. In addition, we develop both a block and an accelerated variant, along with a thorough convergence analysis of these algorithms. A variety of numerical experiments on both synthetic and real-world datasets demonstrate the effectiveness and significant potential of the proposed methods in image and video processing tasks, such as image sequence destriping and video deconvolution.

翻译：张量是表示和分析复杂多维数据的关键工具。随着数据量的持续增长，开发能够直接作用于张量以实现快速有效计算的优化算法需求日益迫切。现实应用中的许多问题可归结为恢复具有稀疏和/或低秩结构的高阶张量的任务。在本工作中，我们提出了新型Kaczmarz算法，结合$\ell_1$范数正则化的威力，通过利用张量数据的稀疏性和/或低秩性来重构高阶张量。此外，我们开发了分块变体与加速变体，并对这些算法进行了全面的收敛性分析。基于合成数据集和真实数据集的多种数值实验表明，所提出方法在图像序列去条带和视频反卷积等图像与视频处理任务中具有显著的有效性和巨大潜力。