mlsanityData to 14 Oct 2026

Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matrices

arXiv 2201.12592 ↗First posted 29 Jan 2022cs.CVcited by 19

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Abstract (arXiv)

It is known that the decomposition in low-rank and sparse matrices (L+S for short) can be achieved by several Robust PCA techniques. Besides the low rankness, the local smoothness (LSS) is a vitally essential prior for many real-world matrix data such as hyperspectral images and surveillance videos, which makes such matrices have low-rankness and local smoothness properties at the same time. This poses an interesting question: Can we make a matrix decomposition in terms of L&LSS +S form exactly? To address this issue, we propose in this paper a new RPCA model based on three-dimensional correlated total variation regularization (3DCTV-RPCA for short) by fully exploiting and encoding the prior expression underlying such joint low-rank and local smoothness matrices. Specifically, using a modification of Golfing scheme, we prove that under some mild assumptions, the proposed 3DCTV-RPCA model can decompose both components exactly, which should be the first theoretical guarantee among all such related methods combining low rankness and local smoothness. In addition, by utilizing Fast Fourier Transform (FFT), we propose an efficient ADMM algorithm with a solid convergence guarantee for solving the resulting optimization problem. Finally, a series of experiments on both simulations and real applications are carried out to demonstrate the general validity of the proposed 3DCTV-RPCA model.

Key facts. Cited by 19 papers in the map (data through 14 Oct 2026). Builds on 8 of its 73 references that are in the map. First posted on arXiv 29 Jan 2022.

Builds on

8 of 73 references.

References · 8

  1. 2018Tensor Robust Principal Component Analysis with A New Tensor Nuclear Normcited by 112
  2. 2017Hyperspectral Image Restoration via Total Variation Regularized Low-rank Tensor Decompositioncited by 61
  3. 2017Multi-channel Weighted Nuclear Norm Minimization for Real Color Image Denoisingcited by 41
  4. 2017Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimizationcited by 35
  5. 2013Incoherence-Optimal Matrix Completioncited by 27
  6. 2017Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recoverycited by 25
  7. 2015Total Variation Regularized Tensor RPCA for Background Subtraction from Compressive Measurementscited by 14
  8. 2017Robust Online Matrix Factorization for Dynamic Background Subtractioncited by 10

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 2; 2023: 3; 2024: 5; 2025: 8; 2026: 1.

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  1. 2022Low-Rank Tensor Function Representation for Multi-Dimensional Data Recoverycited by 46
  2. 2023Guaranteed Tensor Recovery Fused Low-rankness and Smoothnesscited by 41
  3. 2022Fast Noise Removal in Hyperspectral Images via Representative Coefficient Total Variationcited by 14
  4. 2024Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descentcited by 7
  5. 2024Pan-denoising: Guided Hyperspectral Image Denoising via Weighted Represent Coefficient Total Variationcited by 5
  6. 2024Hipandas: Hyperspectral Image Joint Denoising and Super-Resolution by Image Fusion with the Panchromatic Imagecited by 4
  7. 2023Decomposition Ascribed Synergistic Learning for Unified Image Restorationcited by 3
  8. 2025Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theorycited by 2
  9. 2025Hyperspectral Anomaly Detection Fused Unified Nonconvex Tensor Ring Factors Regularizationcited by 2
  10. 2024Haar Nuclear Norms with Applications to Remote Sensing Imagery Restorationcited by 1
  11. 2025Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularizationcited by 1
  12. 2025Kernelized Sparse Fine-Tuning with Bi-level Parameter Competition for Vision Modelscited by 1
7 more
  1. 2023Neural Gradient Regularizercited by 0
  2. 2024Generalized Low-Rank Matrix Completion Model with Overlapping Group Error Representationcited by 0
  3. 2025Score-Based Model for Low-Rank Tensor Recoverycited by 0
  4. 2025Robust Tensor Completion via Gradient Tensor Nulclear L1-L2 Norm for Traffic Data Recoverycited by 0
  5. 2025Beyond Low-rankness: Guaranteed Matrix Recovery via Modified Nuclear Normcited by 0
  6. 2025Exact Recovery of Non-Random Missing Multidimensional Time Series via Temporal Isometric Delay-Embedding Transformcited by 0
  7. 2026Pre-Trained Low-Rank Tensor Decomposition for Multi-Dimensional Image Recoverycited by 0