Abstract
Motivated by the fact that tensor robust principal component analysis (TRPCA) and its variants do not utilize the actual rank value, which limits the recovery performance, and their computational costs are always monumental for large-scale tensor recovery, a fast TRPCA is proposed to recover the low-rank tensor and sparse tensor by estimating the multi-rank vector and adopting Riemannian optimization strategy in this paper. Specifically, a fast multi-rank estimation of low-rank tensor is proposed by modifying the Gershgorin disk theorem-based matrix rank estimation. An innovative TRPCA with Estimated Multi-Rank (TRPCA-EMR) is proposed to eliminate hyperparameter tuning by imposing strict multi-rank equality constraints. Additionally, Riemannian optimization is employed to project each frontal slice of the tensor in Fourier domain onto a low multi-rank manifold in efficiently coping with tensor Singular Value Decomposition (t-SVD) and reduce computational complexity. Experimental results on synthetic and real-world tensor datasets demonstrate TRPCA-EMR’s superior efficiency and effectiveness compared to existing methods, confirming its potential for practical applications and reassuring its reliability.
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The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
References
Candès EJ, Li X, Ma Y, Wright J (2011) Robust principal component analysis? Journal of the ACM (JACM) 58(3):1–37
Xu Z, He R, Xie S, Wu S (2021) Adaptive rank estimate in robust principal component analysis. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 6577–6586
Ma C, Wan M, Xu Y, Ren K, Qian W, Chen Q, Gu G (2022) Infrared target tracking based on proximal robust principal component analysis method. Appl Intell 52(3):2785–2800
Xue N, Deng J, Cheng S, Panagakis Y, Zafeiriou S (2019) Side information for face completion: a robust PCA approach. IEEE Trans Pattern Anal Mach Intell 41(10):2349–2364
Wang P, Wang L, Leung H, Zhang G (2021) Super-resolution mapping based on spatial-spectral correlation for spectral imagery. IEEE Trans Geosci Remote Sens 59(3):2256–2268
Han Z, Wang Y, Zhao Q, Meng D, Lin L, Tang Y et al (2018) A generalized model for robust tensor factorization with noise modeling by mixture of Gaussians. IEEE Trans Neural Netw Learn Syst 29(11):5380–5393
Bahri M, Panagakis Y, Zafeiriou S (2018) Robust Kronecker component analysis. IEEE Trans Pattern Anal Mach Intell 41(10):2365–2379
Zhou Y, Lu H, Cheung Y-M (2019) Probabilistic rank-one tensor analysis with concurrent regularizations. IEEE Trans Cybern 51(7):3496–3509
Shi Q, Cheung Y-M, Lou J (2021) Robust tensor SVD and recovery with rank estimation. IEEE Trans Cybern 52(10):10667–10682
Hillar CJ, Lim L-H (2013) Most tensor problems are NP-hard. Journal of the ACM (JACM) 60(6):1–39
Zhou P, Lu C, Lin Z, Zhang C (2017) Tensor factorization for low-rank tensor completion. IEEE Trans Image Process 27(3):1152–1163
Kilmer ME, Martin CD (2011) Factorization strategies for third-order tensors. Linear Algebra Appl 435(3):641–658
Semerci O, Hao N, Kilmer ME, Miller EL (2014) Tensor-based formulation and nuclear norm regularization for multienergy computed tomography. IEEE Trans Image Process 23(4):1678–1693
Lu C, Feng J, Chen Y, Liu W, Lin Z, Yan S (2016) Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 5249–5257
Lu C, Feng J, Chen Y, Liu W, Lin Z, Yan S (2019) Tensor robust principal component analysis with a new tensor nuclear norm. IEEE Trans Pattern Anal Mach Intell 42(4):925–938
Zhou P, Feng J (2017) Outlier-robust tensor PCA. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2263–2271
Liu Y, Chen L, Zhu C (2018) Improved robust tensor principal component analysis via low-rank core matrix. IEEE J Selected Topics Signal Process 12(6):1378–1389
Zhang F, Wang H, Qin W, Zhao X, Wang J (2023) Generalized nonconvex regularization for tensor rpca and its applications in visual inpainting. Appl Intell 53(20):23124–23146
Jiang T-X, Huang T-Z, Zhao X-L, Deng L-J (2020) Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm. J Comput Appl Math 372:112680
Gao Q, Zhang P, Xia W, Xie D, Gao X, Tao D (2020) Enhanced tensor RPCA and its application. IEEE Trans Pattern Anal Mach Intell 43(6):2133–2140
Qiu H, Wang Y, Tang S, Meng D, Yao Q (2022) Fast and provable nonconvex tensor RPCA. In: International conference on machine learning, pp 18211–18249
Xue J, Zhao Y, Liao W, Chan JC-W (2019) Nonconvex tensor rank minimization and its applications to tensor recovery. Inf Sci 503:109–128
Imbiriba T, Borsoi RA, Bermudez JCM (2020) Low-rank tensor modeling for hyperspectral unmixing accounting for spectral variability. IEEE Trans Geosci Remote Sens 58(3):1833–1842
Yokota T, Lee N, Cichocki A (2017) Robust multilinear tensor rank estimation using higher order singular value decomposition and information criteria. IEEE Trans Signal Process 65(5):1196–1206
Vandereycken B (2013) Low-rank matrix completion by Riemannian optimization. SIAM J Optim 23(2):1214–1236
Wei K, Cai J-F, Chan TF, Leung S (2016) Guarantees of Riemannian optimization for low rank matrix recovery. SIAM J Matrix Anal Appl 37(3):1198–1222
Cai J-F, Li J, Xia D (2023) Generalized low-rank plus sparse tensor estimation by fast Riemannian optimization. J Am Stat Assoc 118(544):2588–2604
Luo Y, Zhang AR (2024) Low-rank tensor estimation via Riemannian Gauss-Newton: statistical optimality and second-order convergence. J Mach Learn Res 24(1):18274–18321
Fang S, Xu Z, Wu S, Xie S (2023) Efficient robust principal component analysis via block Krylov iteration and cur decomposition. In: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp 1348–1357
Wu H-T, Yang J-F, Chen F-K (1995) Source number estimators using transformed gerschgorin radii. IEEE Trans Signal Process 43(6):1325–1333
Zhang Z, Ely G, Aeron S, Hao N, Kilmer M (2014) Novel methods for multilinear data completion and de-noising based on tensor-svd. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 3842–3849
Osher S, Mao Y, Dong B, Yin W (2010) Fast linearized Bregman iteration for compressive sensing and sparse denoising. Commun Math Sci 8(1):93–111
Hale ET, Yin W, Zhang Y (2008) Fixed-point continuation for \(l_1\)-minimization: Methodology and convergence. SIAM J Optim 19(3):1107–1130
Lu C, Feng J, Yan S, Lin Z (2017) A unified alternating direction method of multipliers by majorization minimization. IEEE Trans Pattern Anal Mach Intell 40(3):527–541
Maddalena L, Petrosino A (2015) Towards benchmarking scene background initialization. In: New Trends in Image Analysis and Processing–ICIAP 2015 Workshops: ICIAP 2015 International Workshops, BioFor, CTMR, RHEUMA, ISCA, MADiMa, SBMI, and QoEM, Genoa, Italy, September 7-8, 2015, Proceedings 18, pp 469–476
Bouwmans T, Maddalena L, Petrosino A (2017) Scene background initialization: a taxonomy. Pattern Recogn Lett 96:3–11
Jodoin P-M, Maddalena L, Petrosino A, Wang Y (2017) Extensive benchmark and survey of modeling methods for scene background initialization. IEEE Trans Image Process 26(11):5244–5256
Cao X, Yang L, Guo X (2015) Total variation regularized RPCA for irregularly moving object detection under dynamic background. IEEE Trans Cybern 46(4):1014–1027
Sanches SR, Oliveira C, Sementille AC, Freire V (2019) Challenging situations for background subtraction algorithms. Appl Intell 49:1771–1784
Toyama K, Krumm J, Brumitt B, Meyers B (1999) Wallflower: Principles and practice of background maintenance. In: Proceedings of the seventh IEEE international conference on computer vision, pp 255–261
Sauvalle B, de La Fortelle A (2023) Autoencoder-based background reconstruction and foreground segmentation with background noise estimation. In: IEEE/CVF winter conference on applications of computer vision, pp 3243–3254
Rezaei B, Farnoosh A, Ostadabbas S (2020) G-LBM: generative low-dimensional background model estimation from video sequences. In: European conference on computer vision, pp 293–310
Baumgardner MF, Biehl LL, Landgrebe DA (2015) 220 band aviris hyperspectral image data set: June 12, 1992 indian pine test site 3. Purdue University Res Repository 10(7):991
Acknowledgements
The authors would like to express their sincere gratitude to the Editor and the anonymous reviewers for their invaluable comments and suggestions, which have significantly enhanced the quality and clarity of this paper. Their insightful feedback has been instrumental in refining our research and presenting it more effectively.
Funding
The research leading to these results received funding from the National Nature Science Foundation of China (Grant No.61371190) and the National Nature Science Foundation of China Essential project (Grant No.U2033218).
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Qile Zhu: Conceptualization, Methodology, Validation, Software, Writing - original draft. Shiqian Wu: Formal analysis, Validation, Writing - review & editing. Shun Fang: Writing - review & editing. Qi Wu: Writing - review & editing. Shoulie Xie: Conceptualization, Methodology, Writing - review & editing. Sos Agaian: Supervision, writing - review & editing.
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Zhu, Q., Wu, S., Fang, S. et al. Fast tensor robust principal component analysis with estimated multi-rank and Riemannian optimization. Appl Intell 55, 52 (2025). https://doi.org/10.1007/s10489-024-05899-9
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DOI: https://doi.org/10.1007/s10489-024-05899-9