Tensor Completion via Tensor Networks with a Tucker Wrapper
This work addresses tensor completion for applications like image inpainting and data recovery, but it is incremental as it adapts existing tensor network concepts to this problem.
The paper tackles low-rank tensor completion by proposing a tensor network method with a Tucker wrapper, formulating it as solving nonlinear equations instead of constrained optimization, and demonstrates that the algorithm converges linearly under certain assumptions and performs comparably to state-of-the-art methods in numerical simulations.
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.