Tensor Completion via Tensor Networks with a Tucker Wrapper

TL;DR

This paper introduces a novel tensor network approach with a Tucker wrapper for low-rank tensor completion, formulating it as solving nonlinear equations and demonstrating efficient convergence and competitive performance.

Contribution

It is the first to apply tensor networks with a Tucker wrapper to low-rank tensor completion, offering a new formulation and efficient solution method.

Findings

Method converges linearly to the exact solution under certain conditions.

Algorithm is computationally efficient, dominated by tensor matrix multiplications.

Numerical results show competitive performance with state-of-the-art methods.

Abstract

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.