# Tensor-Ring Nuclear Norm Minimization and Application for Visual Data Completion

arXiv: https://arxiv.org/abs/1903.08888 · HTML: https://mlsanity.com/paper/1903.08888/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Mar 2019.

## Built on by (most cited first, 12 of 12)

- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem](https://mlsanity.com/paper/2110.08754/) (2021)
- [Coarse to Fine Two-Stage Approach to Robust Tensor Completion of Visual Data](https://mlsanity.com/paper/2106.10422/) (2021)
- [Patch Tracking-based Streaming Tensor Ring Completion for Visual Data Recovery](https://mlsanity.com/paper/2105.14620/) (2021)
- [Multi-Tensor Network Representation for High-Order Tensor Completion](https://mlsanity.com/paper/2109.04022/) (2021)
- [Bayesian Robust Tensor Ring Model for Incomplete Multiway Data](https://mlsanity.com/paper/2202.13321/) (2022)
- [A high-order tensor completion algorithm based on Fully-Connected Tensor Network weighted optimization](https://mlsanity.com/paper/2204.01732/) (2022)
- [Scalable and Robust Tensor Ring Decomposition for Large-scale Data](https://mlsanity.com/paper/2305.09044/) (2023)
- [Optical Coherence Tomography Image Enhancement via Block Hankelization and Low Rank Tensor Network Approximation](https://mlsanity.com/paper/2306.11750/) (2023)
- [Quaternion tensor left ring decomposition and application for color image inpainting](https://mlsanity.com/paper/2307.10620/) (2023)
- [Accelerating Large-Scale Regularized High-Order Tensor Recovery](https://mlsanity.com/paper/2506.09594/) (2025)
- [Multi-Dimensional Visual Data Recovery: Scale-Aware Tensor Modeling and Accelerated Randomized Computation](https://mlsanity.com/paper/2602.12982/) (2026)

## Abstract (arXiv)

Tensor ring (TR) decomposition has been successfully used to obtain the state-of-the-art performance in the visual data completion problem. However, the existing TR-based completion methods are severely non-convex and computationally demanding. In addition, the determination of the optimal TR rank is a tough work in practice. To overcome these drawbacks, we first introduce a class of new tensor nuclear norms by using tensor circular unfolding. Then we theoretically establish connection between the rank of the circularly-unfolded matrices and the TR ranks. We also develop an efficient tensor completion algorithm by minimizing the proposed tensor nuclear norm. Extensive experimental results demonstrate that our proposed tensor completion method outperforms the conventional tensor completion methods in the image/video in-painting problem with striped missing values.