# Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm

arXiv: https://arxiv.org/abs/1712.05870 · HTML: https://mlsanity.com/paper/1712.05870/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Dec 2017.

## Built on by (most cited first, 19 of 19)

- [Tensor Recovery Based on A Novel Non-convex Function Minimax Logarithmic Concave Penalty Function](https://mlsanity.com/paper/2206.13506/) (2022)
- [Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent](https://mlsanity.com/paper/2401.11940/) (2024)
- [Low-Rank Tensor Completion via Novel Sparsity-Inducing Regularizers](https://mlsanity.com/paper/2310.06233/) (2023)
- [Bayesian Robust Tensor Ring Model for Incomplete Multiway Data](https://mlsanity.com/paper/2202.13321/) (2022)
- [Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA](https://mlsanity.com/paper/2501.04565/) (2025)
- [Non-convex approaches for low-rank tensor completion under tubal sampling](https://mlsanity.com/paper/2303.12721/) (2023)
- [A Novel Tensor Factorization-Based Method with Robustness to Inaccurate Rank Estimation](https://mlsanity.com/paper/2305.11458/) (2023)
- [Low-rank tensor completion via tensor joint rank with logarithmic composite norm](https://mlsanity.com/paper/2309.16208/) (2023)
- [DTR: A Unified Deep Tensor Representation Framework for Multimedia Data Recovery](https://mlsanity.com/paper/2407.05267/) (2024)
- [Efficient Low-Tubal-Rank Tensor Estimation via Alternating Preconditioned Gradient Descent](https://mlsanity.com/paper/2512.07490/) (2025)
- [Hyperspectral Image Denoising via Multi-modal and Double-weighted Tensor Nuclear Norm](https://mlsanity.com/paper/2101.07681/) (2021)
- [Multi-modal and frequency-weighted tensor nuclear norm for hyperspectral image denoising](https://mlsanity.com/paper/2106.12489/) (2021)
- [Tensor Full Feature Measure and Its Nonconvex Relaxation Applications to Tensor Recovery](https://mlsanity.com/paper/2109.12257/) (2021)
- [Low-Rank Tensor Completion Based on Bivariate Equivalent Minimax-Concave Penalty](https://mlsanity.com/paper/2201.12709/) (2022)
- [Global Weighted Tensor Nuclear Norm for Tensor Robust Principal Component Analysis](https://mlsanity.com/paper/2209.14084/) (2022)
- [Variational Bayesian Inference for Tensor Robust Principal Component Analysis](https://mlsanity.com/paper/2412.18717/) (2024)
- [A novel non-convex minimax p-th order concave penalty function approach to low-rank tensor completion](https://mlsanity.com/paper/2502.19979/) (2025)
- [Outlier-aware Tensor Robust Principal Component Analysis with Self-guided Data Augmentation](https://mlsanity.com/paper/2504.18323/) (2025)
- [Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm](https://mlsanity.com/paper/2606.19046/) (2026)

## Abstract (arXiv)

In this paper, we investigate tensor recovery problems within the tensor singular value decomposition (t-SVD) framework. We propose the partial sum of the tubal nuclear norm (PSTNN) of a tensor. The PSTNN is a surrogate of the tensor tubal multi-rank. We build two PSTNN-based minimization models for two typical tensor recovery problems, i.e., the tensor completion and the tensor principal component analysis. We give two algorithms based on the alternating direction method of multipliers (ADMM) to solve proposed PSTNN-based tensor recovery models. Experimental results on the synthetic data and real-world data reveal the superior of the proposed PSTNN.