# Novel methods for multilinear data completion and de-noising based on tensor-SVD

arXiv: https://arxiv.org/abs/1407.1785 · HTML: https://mlsanity.com/paper/1407.1785/

## Key facts

- Cited by 56 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jul 2014.

## Built on by (most cited first, 30 of 56)

- [Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2212.00262/) (2022)
- [DDS2M: Self-Supervised Denoising Diffusion Spatio-Spectral Model for Hyperspectral Image Restoration](https://mlsanity.com/paper/2303.06682/) (2023)
- [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (2021)
- [Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2105.14320/) (2021)
- [Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion](https://mlsanity.com/paper/2110.08774/) (2021)
- [Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruption](https://mlsanity.com/paper/2305.04080/) (2023)
- [Tensor Decompositions for Hyperspectral Data Processing in Remote Sensing: A Comprehensive Review](https://mlsanity.com/paper/2205.06407/) (2022)
- [Fast Robust Tensor Principal Component Analysis via Fiber CUR Decomposition](https://mlsanity.com/paper/2108.10448/) (2021)
- [Tensor Recovery Based on A Novel Non-convex Function Minimax Logarithmic Concave Penalty Function](https://mlsanity.com/paper/2206.13506/) (2022)
- [A Comparison of Image Denoising Methods](https://mlsanity.com/paper/2304.08990/) (2023)
- [Efficient Tensor Robust PCA under Hybrid Model of Tucker and Tensor Train](https://mlsanity.com/paper/2112.10771/) (2021)
- [Smooth Robust Tensor Completion for Background/Foreground Separation with Missing Pixels: Novel Algorithm with Convergence Guarantee](https://mlsanity.com/paper/2203.16328/) (2022)
- [Robust Implementation of Foreground Extraction and Vessel Segmentation for X-ray Coronary Angiography Image Sequence](https://mlsanity.com/paper/2209.07237/) (2022)
- [OTLRM: Orthogonal Learning-based Low-Rank Metric for Multi-Dimensional Inverse Problems](https://mlsanity.com/paper/2412.11165/) (2024)
- [Non-Convex Tensor Low-Rank Approximation for Infrared Small Target Detection](https://mlsanity.com/paper/2105.14974/) (2021)
- [Image Denoising Using Green Channel Prior](https://mlsanity.com/paper/2408.05923/) (2024)
- [Temporal Network Embedding via Tensor Factorization](https://mlsanity.com/paper/2108.09837/) (2021)
- [Low-Rank Tensor Completion via Novel Sparsity-Inducing Regularizers](https://mlsanity.com/paper/2310.06233/) (2023)
- [Irregular Tensor Low-Rank Representation for Hyperspectral Image Representation](https://mlsanity.com/paper/2410.18388/) (2024)
- [Bayesian Robust Tensor Ring Model for Incomplete Multiway Data](https://mlsanity.com/paper/2202.13321/) (2022)
- [Handling The Non-Smooth Challenge in Tensor SVD: A Multi-Objective Tensor Recovery Framework](https://mlsanity.com/paper/2311.13958/) (2023)
- [Optimal Matrix-Mimetic Tensor Algebras via Variable Projection](https://mlsanity.com/paper/2406.06942/) (2024)
- [Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA](https://mlsanity.com/paper/2501.04565/) (2025)
- [Two New Low Rank Tensor Completion Methods Based on Sum Nuclear Norm](https://mlsanity.com/paper/2108.03002/) (2021)
- [Tensor Factorization via Transformed Tensor-Tensor Product for Image Alignment](https://mlsanity.com/paper/2212.05719/) (2022)
- [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)
- [T-ADAF: Adaptive Data Augmentation Framework for Image Classification Network based on Tensor T-product Operator](https://mlsanity.com/paper/2306.04240/) (2023)
- [Low-rank tensor completion via tensor joint rank with logarithmic composite norm](https://mlsanity.com/paper/2309.16208/) (2023)
- [Coseparable Nonnegative Tensor Factorization With T-CUR Decomposition](https://mlsanity.com/paper/2401.16836/) (2024)

## Abstract (arXiv)

In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].