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

arXiv: https://arxiv.org/abs/2309.16208 · HTML: https://mlsanity.com/paper/2309.16208/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 50 references that are in the map.
- First posted on arXiv 28 Sep 2023.

## Builds on (most cited first, 10 of 10)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
- [Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion](https://mlsanity.com/paper/1909.06982/) (2019)
- [Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm](https://mlsanity.com/paper/1712.05870/) (2017)
- [Tensor N-tubal rank and its convex relaxation for low-rank tensor recovery](https://mlsanity.com/paper/1812.00688/) (2018)
- [Dictionary Learning with Low-rank Coding Coefficients for Tensor Completion](https://mlsanity.com/paper/2009.12507/) (2020)
- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [Tensor Recovery Based on A Novel Non-convex Function Minimax Logarithmic Concave Penalty Function](https://mlsanity.com/paper/2206.13506/) (2022)

## Built on by (most cited first, 1 of 1)

- [Superpixel-informed Continuous Low-Rank Tensor Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2508.12261/) (2025)

## Abstract (arXiv)

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.