mlsanityData to 7 Oct 2026

Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm

arXiv 1804.03728 ↗First posted 10 Apr 2018stat.ML cs.LGcited by 110

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Abstract (arXiv)

In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product). Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.

Key facts. Cited by 110 papers in the map (data through 7 Oct 2026). First posted on arXiv 10 Apr 2018.

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Citing papers per year: 2021: 18; 2022: 19; 2023: 23; 2024: 15; 2025: 24; 2026: 11.

110 papers.

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  1. 2022Low-Rank Tensor Function Representation for Multi-Dimensional Data Recoverycited by 45
  2. 2023Guaranteed Tensor Recovery Fused Low-rankness and Smoothnesscited by 41
  3. 2024HIR-Diff: Unsupervised Hyperspectral Image Restoration Via Improved Diffusion Modelscited by 38
  4. 2021T-SVDNet: Exploring High-Order Prototypical Correlations for Multi-Source Domain Adaptationcited by 24
  5. 2021Low-Rank Hankel Tensor Completion for Traffic Speed Estimationcited by 22
  6. 2022Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matricescited by 19
  7. 2022Laplacian Convolutional Representation for Traffic Time Series Imputationcited by 18
  8. 2022Correlating sparse sensing for large-scale traffic speed estimation: A Laplacian-enhanced low-rank tensor kriging approachcited by 17
  9. 2021Self-Supervised Nonlinear Transform-Based Tensor Nuclear Norm for Multi-Dimensional Image Recoverycited by 16
  10. 2022Fast Noise Removal in Hyperspectral Images via Representative Coefficient Total Variationcited by 13
  11. 2023Domain Generalization via Nuclear Norm Regularizationcited by 12
  12. 2022Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descentcited by 11
38 more
  1. 2023Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruptioncited by 11
  2. 2021Fast Robust Tensor Principal Component Analysis via Fiber CUR Decompositioncited by 8
  3. 2022Semi-Supervised Subspace Clustering via Tensor Low-Rank Representationcited by 7
  4. 2022Tensor Recovery Based on A Novel Non-convex Function Minimax Logarithmic Concave Penalty Functioncited by 7
  5. 2022Deep Unfolded Tensor Robust PCA with Self-supervised Learningcited by 7
  6. 2023Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximationcited by 7
  7. 2024Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descentcited by 7
  8. 2024Clients Collaborate: Flexible Differentially Private Federated Learning with Guaranteed Improvement of Utility-Privacy Trade-offcited by 6
  9. 2021Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problemcited by 5
  10. 2024OTLRM: Orthogonal Learning-based Low-Rank Metric for Multi-Dimensional Inverse Problemscited by 5
  11. 2021Multi-Slice Low-Rank Tensor Decomposition Based Multi-Atlas Segmentation: Application to Automatic Pathological Liver CT Segmentationcited by 4
  12. 2021Non-Convex Tensor Low-Rank Approximation for Infrared Small Target Detectioncited by 4
  13. 2021Hankel-structured Tensor Robust PCA for Multivariate Traffic Time Series Anomaly Detectioncited by 4
  14. 2022T2LR-Net: An unrolling network learning transformed tensor low-rank prior for dynamic MR image reconstructioncited by 4
  15. 2022Robust Implementation of Foreground Extraction and Vessel Segmentation for X-ray Coronary Angiography Image Sequencecited by 4
  16. 2023Transformed Low-Rank Parameterization Can Help Robust Generalization for Tensor Neural Networkscited by 4
  17. 2023Semantically Consistent Multi-view Representation Learningcited by 4
  18. 2023Learning Spatial-Temporal Regularized Tensor Sparse RPCA for Background Subtractioncited by 4
  19. 2026HADAR-Based Thermal Infrared Hyperspectral Image Restorationcited by 4
  20. 2021Streaming data preprocessing via online tensor recovery for large environmental sensor networkscited by 3
  21. 2023H2TF for Hyperspectral Image Denoising: Where Hierarchical Nonlinear Transform Meets Hierarchical Matrix Factorizationcited by 3
  22. 2023Low-Rank Tensor Completion via Novel Sparsity-Inducing Regularizerscited by 3
  23. 2024S^2MVTC: a Simple yet Efficient Scalable Multi-View Tensor Clusteringcited by 3
  24. 2024Irregular Tensor Low-Rank Representation for Hyperspectral Image Representationcited by 3
  25. 2021Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximationcited by 2
  26. 2021Multi-Tensor Network Representation for High-Order Tensor Completioncited by 2
  27. 2022Dynamic Cardiac MRI Reconstruction Using Combined Tensor Nuclear Norm and Casorati Matrix Nuclear Norm Regularizationscited by 2
  28. 2022Dynamic MRI using Learned Transform-based Tensor Low-Rank Network (LT2LR-Net)cited by 2
  29. 2023Personalized Tucker Decomposition: Modeling Commonality and Peculiarity on Tensor Datacited by 2
  30. 2023Handling The Non-Smooth Challenge in Tensor SVD: A Multi-Objective Tensor Recovery Frameworkcited by 2
  31. 2024Semi-supervised Symmetric Non-negative Matrix Factorization with Low-Rank Tensor Representationcited by 2
  32. 2025Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCAcited by 2
  33. 2025Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theorycited by 2
  34. 2021Two New Low Rank Tensor Completion Methods Based on Sum Nuclear Normcited by 1
  35. 2021Effective Streaming Low-tubal-rank Tensor Approximation via Frequent Directionscited by 1
  36. 2021Ergodic Limits, Relaxations, and Geometric Properties of Random Walk Node Embeddingscited by 1
  37. 2022Working memory inspired hierarchical video decomposition with transformative representationscited by 1
  38. 2022Low-rank Meets Sparseness: An Integrated Spatial-Spectral Total Variation Approach to Hyperspectral Denoisingcited by 1

50 most cited of 110 All 110 (JSON)