  Tensor recovery has recently arisen in a lot of application fields, such as
transportation, medical imaging and remote sensing. Under the assumption that
signals possess sparse and/or low-rank structures, many tensor recovery methods
have been developed to apply various regularization techniques together with
the operator-splitting type of algorithms. Due to the unprecedented growth of
data, it becomes increasingly desirable to use streamlined algorithms to
achieve real-time computation, such as stochastic optimization algorithms that
have recently emerged as an efficient family of methods in machine learning. In
this work, we propose a novel algorithmic framework based on the Kaczmarz
algorithm for tensor recovery. We provide thorough convergence analysis and its
applications from the vector case to the tensor one. Numerical results on a
variety of tensor recovery applications, including sparse signal recovery,
low-rank tensor recovery, image inpainting and deconvolution, illustrate the
enormous potential of the proposed methods.
