  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.
