# Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization

arXiv: https://arxiv.org/abs/1708.04181 · HTML: https://mlsanity.com/paper/1708.04181/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Aug 2017.

## Built on by (most cited first, 30 of 35)

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- [Theoretical Constraints on the Expressive Power of mathsfRoPE-based Tensor Attention Transformers](https://mlsanity.com/paper/2412.18040/) (2024)
- [Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descent](https://mlsanity.com/paper/2206.09109/) (2022)
- [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)
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- [Get your Foes Fooled: Proximal Gradient Split Learning for Defense against Model Inversion Attacks on IoMT data](https://mlsanity.com/paper/2201.04569/) (2022)
- [Deep Unfolded Tensor Robust PCA with Self-supervised Learning](https://mlsanity.com/paper/2212.11346/) (2022)
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- [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)
- [Multi-Slice Low-Rank Tensor Decomposition Based Multi-Atlas Segmentation: Application to Automatic Pathological Liver CT Segmentation](https://mlsanity.com/paper/2102.12056/) (2021)
- [Non-Convex Tensor Low-Rank Approximation for Infrared Small Target Detection](https://mlsanity.com/paper/2105.14974/) (2021)
- [Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning](https://mlsanity.com/paper/2203.02189/) (2022)
- [Learning Spatial-Temporal Regularized Tensor Sparse RPCA for Background Subtraction](https://mlsanity.com/paper/2309.15576/) (2023)
- [Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA](https://mlsanity.com/paper/2501.04565/) (2025)
- [Time and Memory Trade-off of KV-Cache Compression in Tensor Transformer Decoding](https://mlsanity.com/paper/2503.11108/) (2025)
- [Effective Streaming Low-tubal-rank Tensor Approximation via Frequent Directions](https://mlsanity.com/paper/2108.10129/) (2021)
- [Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularization](https://mlsanity.com/paper/2205.03059/) (2022)
- [IncSAR: A Dual Fusion Incremental Learning Framework for SAR Target Recognition](https://mlsanity.com/paper/2410.05820/) (2024)
- [Knowledge Graph Completion by Intermediate Variables Regularization](https://mlsanity.com/paper/2506.02749/) (2025)
- ["Sparse + Low-Rank'' Tensor Completion Approach for Recovering Images and Videos](https://mlsanity.com/paper/2110.09298/) (2021)
- [Highlight Specular Reflection Separation based on Tensor Low-rank and Sparse Decomposition Using Polarimetric Cues](https://mlsanity.com/paper/2207.03543/) (2022)
- [Infrared Small Target Detection Using Double-Weighted Multi-Granularity Patch Tensor Model With Tensor-Train Decomposition](https://mlsanity.com/paper/2310.05347/) (2023)
- [Low-Multi-Rank High-Order Bayesian Robust Tensor Factorization](https://mlsanity.com/paper/2311.05888/) (2023)

## Abstract (arXiv)

This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA (Candes et al. 2011) to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) (Kilmer and Martin 2011) and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor X∈ℝn1× n2× n3 such that X=L0+E0, where L0 has low tubal rank and E0 is sparse. Is that possible to recover both components? In this work, we prove that under certain suitable assumptions, we can recover both the low-rank and the sparse components exactly by simply solving a convex program whose objective is a weighted combination of the tensor nuclear norm and the ℓ1-norm, i.e., min_L, E  |L|_*+λ|E|1,  s.t.  X=L+E, where λ= 1/√max(n1,n2)n3. Interestingly, TRPCA involves RPCA as a special case when n3=1 and thus it is a simple and elegant tensor extension of RPCA. Also numerical experiments verify our theory and the application for the image denoising demonstrates the effectiveness of our method.