Flexible robust principal component analysis

  • Zinan He
  • Jigang WuEmail author
  • Na Han
Original Article


The error correction problem is a very important topic in machine learning. However, existing methods only focus on data recovery and ignore data compact representation. In this paper, we propose a flexible robust principal component analysis (FRPCA) method in which two different matrices are used to perform error correction and the data compact representation can be obtained by using one of matrices. Moreover, FRPCA selects the most relevant features to guarantee that the recovered data can faithfully preserve the original data semantics. The learning is done by solving a nuclear-norm regularized minimization problem, which is convex and can be solved in polynomial time. Experiments were conducted on image sequences containing targets of interest in a variety of environments, e.g., offices, campuses. We also compare our method with existing method in recovering the face images from corruptions. Experimental results show that the proposed method achieves better performances and it is more practical than the existing approaches.


Error correction Robust principal component analysis (RPCA) Subspace learning 



This work was supported in part by the National Key R & D Program of China under Grant No. 2018YFB1003201, the National Natural Science Foundation of China under Grants 61672171 and 61772141, the Guangdong Natural Science foundation under Grants 2018B030311007, and the Major project of Educational Commission of Guangdong Province under Grants 2016KZDXM052. The corresponding author is Jigang Wu (


  1. 1.
    Costeira JP, Kanade T (1998) A multibody factorization method for independently moving objects. Int J Comput Vis 29(3):159–179CrossRefGoogle Scholar
  2. 2.
    Belhumeur P, Hespanha J, Kriegman D (2002) Eigenfaces versus fisherfaces: recognition using class specific linear projection. IEEE Trans Pattern Anal Mach Intell 19(7):711–720CrossRefGoogle Scholar
  3. 3.
    De la Torre F, Black MJ (2001) Robust principal component analysis for computer vision. In: Proceedings 8th IEEE international conference on computer vision, ICCV 2001. IEEE, 2001, 1, Barcelona, Spain, pp 362–369Google Scholar
  4. 4.
    Eldar YC, Mishali M (2009) Robust recovery of signals from a structured union of subspaces. IEEE Trans Inf Theory 55(11):5302–5316MathSciNetCrossRefGoogle Scholar
  5. 5.
    Elhamifar E, Vidal R (2009) Sparse subspace clustering. In: Proceedings of the IEEE conference on computer vision and pattern recoginition, vol 2, pp 2790–2797Google Scholar
  6. 6.
    Liu G, Lin Z, Yu Y (2010) Robust subspace segmentation by low-rank representation. In: Proceedings of the international conference on machine learning, pp 1–8Google Scholar
  7. 7.
    Li S, Fu Y (2014) Robust subspace discovery through supervised low-rank constraints. In: Proceedings of the SIAM international conference on data mining, pp 163–171Google Scholar
  8. 8.
    Guyon I, Elisseeff A (2003) An introduction to variable and feature selection. J Mach Learn Res 3:1157–1182zbMATHGoogle Scholar
  9. 9.
    Kim E, Lee M, Oh S (2015) Elastic-net regularization of singular values for robust subspace learning. In: Proceedings of the IEEE conference of computer vision and pattern recognition, pp 915–923Google Scholar
  10. 10.
    Wen J et al (2019) Robust sparse linear discriminant analysis. IEEE Trans Circ Syst Video Technol 29(2):390–403CrossRefGoogle Scholar
  11. 11.
    Wen J, Zhong Z, Zhang Z, Fei L, Lai Z, Chen R (2018) Adaptive locality preserving regression. IEEE Trans Circuits Syst Video Technol. CrossRefGoogle Scholar
  12. 12.
    Fei L et al (2017) Low rank representation with adaptive distance penalty for semi-supervised subspace classification. Pattern Recogn 67(2017):252–262CrossRefGoogle Scholar
  13. 13.
    Fang X, Han N, Jigang W, Xu Y, Yang J, Wong WK, Li X (2018) Approximate low-rank projection learning for feature extraction. IEEE Trans Neural Netw Learn Syst 29(11):5228–5241MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wen J, Han N, Fang X, Fei L, Yan K, Zhan S (2019) Low-rank preserving projection via graph regularized reconstruction. IEEE Trans Cybern 49(4):1279–1291CrossRefGoogle Scholar
  15. 15.
    Pearson K (1901) On lines and planes of closest fit to systems of points in space. Philos Magn 2(6):559–572CrossRefGoogle Scholar
  16. 16.
    Belhumeur PN, Hespanha JP, Kriegman DJ (1997) Eigenfaces vs. fisherfaces: recognition using class specific linear projection. IEEE Trans Pattern Anal Mach Intell 19(7):711–720CrossRefGoogle Scholar
  17. 17.
    He X, Niyogi P (2004) Locality preserving projections. In: Advances in neural information processing systems pp 153–160Google Scholar
  18. 18.
    He X et al (2005) Neighborhood preserving embedding. Computer Vision, 2005. ICCV 2005. In: 10th IEEE international conference on Vol 2, IEEEGoogle Scholar
  19. 19.
    Yan S et al (2007) Graph embedding and extensions: a general framework for dimensionality reduction. IEEE Trans Pattern Anal Mach Intell 29(1):40–51CrossRefGoogle Scholar
  20. 20.
    Zou H, Hastie T, Tibshirani R (2006) Sparse principal component analysis. J Comput Graph Stat 15(2):265–286MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lai Z, Yong X, Yang J, Shen L, Zhang D (2017) Rotational invariant dimensionality reduction algorithms. IEEE Trans Cybern 47(11):3733–3746CrossRefGoogle Scholar
  22. 22.
    Lai Z, Yong X, Yang J, Tang J, Zhang D (2013) Sparse tensor discriminant analysis. IEEE Trans Image Process 22(10):3904–3915MathSciNetCrossRefGoogle Scholar
  23. 23.
    Cai D, He X, Han J (2007) Spectral regression: a unified approach for sparse subspace learning. In: 7th IEEE international conference on data mining (ICDM 2007). IEEEGoogle Scholar
  24. 24.
    Candès EJ et al (2011) Robust principal component analysis? J ACM (JACM) 58(3):11MathSciNetCrossRefGoogle Scholar
  25. 25.
    Maronna RA (1976) Robust M-estimators of multivariate location and scatter. Ann Stat 4(1):51–67MathSciNetCrossRefGoogle Scholar
  26. 26.
    Devlin SJ, Gnanadesikan R, Kettenring JR (1981) Robust estimation of dispersion matrices and principal components. J Am Stat Assoc 76(374):354–362CrossRefGoogle Scholar
  27. 27.
    Li G, Chen Z (1985) Projection-pursuit approach to robust dispersion matrices and principal components: primary theory and Monte Carlo. J Am Stat Assoc 80(391):759–766CrossRefGoogle Scholar
  28. 28.
    Torre F, Black M (2001) Robust principal component analysis for computer vision. In: Proceedings of the international conference computer vision, Vancouver, BC, pp 362–369Google Scholar
  29. 29.
    Wright J, Ganesh A, Rao S, Peng Y, Ma Y (2009) Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization. In: Proceedings of neural information processing system, pp 1–9Google Scholar
  30. 30.
    Xu H, Caramanis C, Sanghavi S (2010) Robust PCA via outlier pursuit. In: Proceedings of advances in neural information processing system, pp 1–30Google Scholar
  31. 31.
    Ma Y, Derksen H, Hong W, Wright J (2007) Segmentation of multivariate mixed data via lossy data coding and compression. IEEE Trans Pattern Anal Mach Intell 29(9):1546–1562CrossRefGoogle Scholar
  32. 32.
    Bao B-K et al (2012) Inductive robust principal component analysis. IEEE Trans Image Process 21(8):3794–3800MathSciNetCrossRefGoogle Scholar
  33. 33.
    Chung FRK (1997) Spectral graph theory. AMS, Providence, RIzbMATHGoogle Scholar
  34. 34.
    Donoho DL (2006) For most large underdetermined systems of equations, the minimal \(\ell ^{1}\)-norm near-solution approximates the sparsest nearsolution. Commun Pure Appl Math 59(7):907–934CrossRefGoogle Scholar
  35. 35.
    Lin Z, Chen M, Wu L, Ma Y (2009) The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. Department of Electrical and Computer Engineering, UIUC, Urbana, Technical Report, UILU-ENG-09-2215Google Scholar
  36. 36.
    Cai X, Ding C, Nie F, Huang H (2013) The equivalent of low-rank linear regressions and linear discriminant analysis based regressions. In: Proceedings of the 19th ACM SIGKDD conference knowledge discovery data mining, pp 1124–1132Google Scholar
  37. 37.
    Fang X, Xu Y, Li X, Lai Z, Wong WK (2016) Robust semi-supervised subspace clustering via non-negative low-rank representation. IEEE Trans Cybern 46(8):1828–1838CrossRefGoogle Scholar
  38. 38.
    Liu G, Lin Z, Yan S, Sun J, Yu Y, Ma Y (2013) Robust recovery of subspace structures by low-rank representation. IEEE Trans Pattern Anal Mach Intell 35(1):171–184CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyGuangdong University of TechnologyGuangzhouChina

Personalised recommendations