  In recent years, low-rank tensor completion (LRTC) has received considerable
attention due to its applications in image/video inpainting, hyperspectral data
recovery, etc. With different notions of tensor rank (e.g., CP, Tucker, tensor
train/ring, etc.), various optimization based numerical methods are proposed to
LRTC. However, tensor network based methods have not been proposed yet. In this
paper, we propose to solve LRTC via tensor networks with a Tucker wrapper. Here
by "Tucker wrapper" we mean that the outermost factor matrices of the tensor
network are all orthonormal. We formulate LRTC as a problem of solving a system
of nonlinear equations, rather than a constrained optimization problem. A
two-level alternative least square method is then employed to update the
unknown factors. The computation of the method is dominated by tensor matrix
multiplications and can be efficiently performed. Also, under proper
assumptions, it is shown that with high probability, the method converges to
the exact solution at a linear rate. Numerical simulations show that the
proposed algorithm is comparable with state-of-the-art methods.
