mlsanityData to 7 Oct 2026

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

arXiv 1708.04181 ↗First posted 14 Aug 2017cs.CVcited by 35

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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.

Key facts. Cited by 35 papers in the map (data through 7 Oct 2026). First posted on arXiv 14 Aug 2017.

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Citing papers per year: 2021: 4; 2022: 13; 2023: 4; 2024: 5; 2025: 6; 2026: 3.

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  1. 2024Training Tensor Attention Efficiently: From Cubic to Almost Linear Timecited by 29
  2. 2022Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matricescited by 19
  3. 2022Robust and Explainable Autoencoders for Unsupervised Time Series Outlier Detection---Extended Versioncited by 19
  4. 2022Correlating sparse sensing for large-scale traffic speed estimation: A Laplacian-enhanced low-rank tensor kriging approachcited by 17
  5. 2022Fast Noise Removal in Hyperspectral Images via Representative Coefficient Total Variationcited by 13
  6. 2024Theoretical Constraints on the Expressive Power of mathsfRoPE-based Tensor Attention Transformerscited by 12
  7. 2022Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descentcited by 11
  8. 2023Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruptioncited by 11
  9. 2022Tensor Decompositions for Hyperspectral Data Processing in Remote Sensing: A Comprehensive Reviewcited by 9
  10. 2025Speckle2Self: Self-Supervised Ultrasound Speckle Reduction Without Clean Datacited by 9
  11. 2024Long-range Turbulence Mitigation: A Large-scale Dataset and A Coarse-to-fine Frameworkcited by 8
  12. 2022Get your Foes Fooled: Proximal Gradient Split Learning for Defense against Model Inversion Attacks on IoMT datacited by 7
23 more
  1. 2022Deep Unfolded Tensor Robust PCA with Self-supervised Learningcited by 7
  2. 2024Clients Collaborate: Flexible Differentially Private Federated Learning with Guaranteed Improvement of Utility-Privacy Trade-offcited by 6
  3. 2022Smooth Robust Tensor Completion for Background/Foreground Separation with Missing Pixels: Novel Algorithm with Convergence Guaranteecited by 5
  4. 2021Multi-Slice Low-Rank Tensor Decomposition Based Multi-Atlas Segmentation: Application to Automatic Pathological Liver CT Segmentationcited by 4
  5. 2021Non-Convex Tensor Low-Rank Approximation for Infrared Small Target Detectioncited by 4
  6. 2022Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learningcited by 4
  7. 2022Robust Implementation of Foreground Extraction and Vessel Segmentation for X-ray Coronary Angiography Image Sequencecited by 4
  8. 2023Learning Spatial-Temporal Regularized Tensor Sparse RPCA for Background Subtractioncited by 4
  9. 2025Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCAcited by 2
  10. 2025Time and Memory Trade-off of KV-Cache Compression in Tensor Transformer Decodingcited by 2
  11. 2021Effective Streaming Low-tubal-rank Tensor Approximation via Frequent Directionscited by 1
  12. 2022Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularizationcited by 1
  13. 2024IncSAR: A Dual Fusion Incremental Learning Framework for SAR Target Recognitioncited by 1
  14. 2025Knowledge Graph Completion by Intermediate Variables Regularizationcited by 1
  15. 2021"Sparse + Low-Rank'' Tensor Completion Approach for Recovering Images and Videoscited by 0
  16. 2022Highlight Specular Reflection Separation based on Tensor Low-rank and Sparse Decomposition Using Polarimetric Cuescited by 0
  17. 2023Infrared Small Target Detection Using Double-Weighted Multi-Granularity Patch Tensor Model With Tensor-Train Decompositioncited by 0
  18. 2023Low-Multi-Rank High-Order Bayesian Robust Tensor Factorizationcited by 0
  19. 2025Triply Laplacian Scale Mixture Modeling for Seismic Data Noise Suppressioncited by 0
  20. 2025Outlier-aware Tensor Robust Principal Component Analysis with Self-guided Data Augmentationcited by 0
  21. 2026Heavy-Tailed Principal Component Analysiscited by 0
  22. 2026Breaking the Periodicity Assumption: Robust Tensorial Multi-View Clustering via Graph-Spectral Low-Rank Learningcited by 0
  23. 2026Pre-Trained Low-Rank Tensor Decomposition for Multi-Dimensional Image Recoverycited by 0