# "Sparse + Low-Rank'' Tensor Completion Approach for Recovering Images and Videos

arXiv: https://arxiv.org/abs/2110.09298 · HTML: https://mlsanity.com/paper/2110.09298/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 61 references that are in the map.
- First posted on arXiv 18 Oct 2021.

## Builds on (most cited first, 15 of 15)

- [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)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Tensor Completion Algorithms in Big Data Analytics](https://mlsanity.com/paper/1711.10105/) (2017)
- [Framelet Representation of Tensor Nuclear Norm for Third-Order Tensor Completion](https://mlsanity.com/paper/1909.06982/) (2019)
- [Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization](https://mlsanity.com/paper/1708.04181/) (2017)
- [Smooth PARAFAC Decomposition for Tensor Completion](https://mlsanity.com/paper/1505.06611/) (2015)
- [On Tensor Completion via Nuclear Norm Minimization](https://mlsanity.com/paper/1405.1773/) (2014)
- [Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm](https://mlsanity.com/paper/1510.06895/) (2015)
- [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)
- [Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements](https://mlsanity.com/paper/1806.02511/) (2018)
- [Deep Plug-and-play Prior for Low-rank Tensor Completion](https://mlsanity.com/paper/1905.04449/) (2019)
- [Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications](https://mlsanity.com/paper/1503.01444/) (2015)

## Abstract (arXiv)

Recovering color images and videos from highly undersampled data is a fundamental and challenging task in face recognition and computer vision. By the multi-dimensional nature of color images and videos, in this paper, we propose a novel tensor completion approach, which is able to efficiently explore the sparsity of tensor data under the discrete cosine transform (DCT). Specifically, we introduce two ``sparse + low-rank'' tensor completion models as well as two implementable algorithms for finding their solutions. The first one is a DCT-based sparse plus weighted nuclear norm induced low-rank minimization model. The second one is a DCT-based sparse plus p-shrinking mapping induced low-rank optimization model. Moreover, we accordingly propose two implementable augmented Lagrangian-based algorithms for solving the underlying optimization models. A series of numerical experiments including color image inpainting and video data recovery demonstrate that our proposed approach performs better than many existing state-of-the-art tensor completion methods, especially for the case when the ratio of missing data is high.