Tensor completion using patch-wise high order Hankelization and randomized tensor ring initialization

Abstract

Recently, tensor completion (TC), namely high-order extension of matrix completion, has aroused widespread attention. Many priors have been investigated turning the ill-posed TC problem into a well-posed one. The typical methods, however, either perform suboptimally due to missing consideration of appropriate priors, or suffer from overly complicated models by combining too many constraints. In this study, we go deeper by providing a detailed analysis on the rank properties in patch-wise and Hankel cases, figuring out both these two ingredients ensure stricter low-rank prior. This prompts us to design a resultful yet simple TC method using patch-wise and multi-way tensor extension. Specifically, strict low-rank property can be guaranteed in the formulated tensors and this is achieved without requiring any neighborhood setting. Moreover, to approximate the intrinsic rank of the Hankelized tensor, tensor ring decomposition is introduced after deeply comparing its superiority over other high-order decomposition techniques. The proposed method can be efficiently solved using alternating least squares (ALS) approach. Furthermore, a randomized tensor ring approximation is presented for fast initialization. Experimental results on color image inpainting demonstrate that the proposed method can generate better recovery performance compared with state-of-the-art low-rank related competitors, especially in cases of quite limited known pixels, e.g., 10%.