# Low-Rank Tensor Completion by Truncated Nuclear Norm Regularization

arXiv: https://arxiv.org/abs/1712.00704 · HTML: https://mlsanity.com/paper/1712.00704/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2017.

## Built on by (most cited first, 7 of 7)

- [Tensor Completion with Provable Consistency and Fairness Guarantees for Recommender Systems](https://mlsanity.com/paper/2204.01815/) (2022)
- [Learning Low-Rank Feature for Thorax Disease Classification](https://mlsanity.com/paper/2404.18933/) (2024)
- [Diffusion on Graph: Augmentation of Graph Structure for Node Classification](https://mlsanity.com/paper/2503.12563/) (2025)
- [Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularization](https://mlsanity.com/paper/2205.03059/) (2022)
- [Image Reconstruction using Superpixel Clustering and Tensor Completion](https://mlsanity.com/paper/2305.09564/) (2023)
- [An Admissible Shift-Consistent Method for Recommender Systems](https://mlsanity.com/paper/2307.08857/) (2023)
- [Color Image Recovery Using Generalized Matrix Completion over Higher-Order Finite Dimensional Algebra](https://mlsanity.com/paper/2308.02621/) (2023)

## Abstract (arXiv)

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.