Low-rank tensor completion via tensor joint rank with logarithmic composite norm

TL;DR

This paper introduces a novel tensor completion method that leverages joint low-rank structures and a logarithmic composite norm to improve recovery performance, especially with very limited observed data.

Contribution

The paper proposes the TJLC method that combines Tucker and tubal ranks with a new logarithmic composite norm, along with theoretical convergence guarantees.

Findings

Outperforms state-of-the-art methods on real datasets.

Achieves accurate recovery with as low as 1% observed data.

Significantly improves recovery as observed data increases.

Abstract

Low-rank tensor completion (LRTC) aims to recover a complete low-rank tensor from incomplete observed tensor, attracting extensive attention in various practical applications such as image processing and computer vision. However, current methods often perform well only when there is a sufficient of observed information, and they perform poorly or may fail when the observed information is less than 5\%. In order to improve the utilization of observed information, a new method called the tensor joint rank with logarithmic composite norm (TJLC) method is proposed. This method simultaneously exploits two types of tensor low-rank structures, namely tensor Tucker rank and tubal rank, thereby enhancing the inherent correlations between known and missing elements. To address the challenge of applying two tensor ranks with significantly different directly to LRTC, a new tensor Logarithmic composite norm is further proposed. Subsequently, the TJLC model and algorithm for the LRTC problem are proposed. Additionally, theoretical convergence guarantees for the TJLC method are provided. Experiments on various real datasets demonstrate that the proposed method outperforms state-of-the-art methods significantly. Particularly, the proposed method achieves satisfactory recovery even when the observed information is as low as 1\%, and the recovery performance improves significantly as the observed information increases.