Low-Rank Tensor Completion by Truncated Nuclear Norm Regularization

TL;DR

This paper introduces a novel tensor truncated nuclear norm (T-TNN) method for low-rank tensor completion, improving the accuracy of recovering incomplete visual data by better approximating tensor rank.

Contribution

It extends the truncated nuclear norm to tensors, providing a new definition and optimization approach that enhances low-rank tensor completion performance.

Findings

Outperforms existing tensor completion methods on real-world data

Utilizes tensor singular value decomposition and ADMM for optimization

Achieves superior recovery accuracy in experiments

Abstract

Currently, low-rank tensor completion has gained cumulative attention in recovering incomplete visual data whose partial elements are missing. By taking a color image or video as a three-dimensional (3D) tensor, previous studies have suggested several definitions of tensor nuclear norm. However, they have limitations and may not properly approximate the real rank of a tensor. Besides, they do not explicitly use the low-rank property in optimization. It is proved that the recently proposed truncated nuclear norm (TNN) can replace the traditional nuclear norm, as a better estimation to the rank of a matrix. Thus, this paper presents a new method called the tensor truncated nuclear norm (T-TNN), which proposes a new definition of tensor nuclear norm and extends the truncated nuclear norm from the matrix case to the tensor case. Beneficial from the low rankness of TNN, our approach improves the efficacy of tensor completion. We exploit the previously proposed tensor singular value decomposition and the alternating direction method of multipliers in optimization. Extensive experiments on real-world videos and images demonstrate that the performance of our approach is superior to those of existing methods.