The Robust Sparse PCA for Data Reconstructive via Weighted Elastic Net

Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 202)


2 1-norm is widely used to measure coding residual in principal component analysis (PCA). In this case, it usually assumes that the residual follows Gaussian/Laplacian distribution. However, it may fail to describe the coding errors in practice when there are outliers. Toward this end, this paper proposes a robust sparse PCA (RSPCA) approach to solve the outlier problem, by modeling the sparse coding as a sparsity-constrained weighted regression problem. By using a series of equivalent transformations, we show RSPCA is equivalent to the weighted elastic net (WEN) problem and thus the least angle regression elastic net (LARS-EN) method is used to yield the optimal solution. Simulation results illustrated the effectiveness of this approach.


Robust statistics Principal component analysis Sparse representation Elastic net 



This work was supported by “the Fundamental Research Funds for the Central Universities” under award number ZYGX2010J016.


  1. 1.
    Aanæs H, Fisker R, Astrom K, Carstensen JM (2002) Robust factorization. IEEE Trans Pattern Anal Mach Intell 24(9):1215–1225Google Scholar
  2. 2.
    Baccini A, Besse P, and Falguerolles de A (1996), An l1-norm pca and a heuristic approach. Ordinal and symbolic data analysis, pp 359–368Google Scholar
  3. 3.
    Candes EJ, Li X, Ma Y, Wright J (2009) Robust principal component analysis? Arxiv preprint ArXiv:0912.3599Google Scholar
  4. 4.
    Croux C, Filzmoser P, Fritz H (2011) Robust sparse principal component analysis. Catholic University of Leuven Department of Decision Science and Information Management Working Paper No. 1113Google Scholar
  5. 5.
    d’Aspremont v, El Ghaoui L, Jordan MI, Lanckriet GRG (2004) A direct formulation for sparse PCA using semidefinite programming. Computer Science Division, University of CaliforniaGoogle Scholar
  6. 6.
    De la Torre F, Black MJ (2001) Robust principal component analysis for computer vision. In: IEEE international conference on computer vision (ICCV), vol 1. IEEE, pp 362–369Google Scholar
  7. 7.
    De La Torre F, Black MJ (2003) A framework for robust subspace learning. Int J Comput Vis 54(1):117–142Google Scholar
  8. 8.
    Ding C, Zhou D, He X, Zha H (2006) R 1-pca: rotational invariant l 1-norm principal component analysis for robust subspace factorization. In: Proceedings of the 23rd international conference on machine learning. ACM, New York, pp 281–288Google Scholar
  9. 9.
    Frieze A, Kannan R, Vempala S (2004) Fast monte-carlo algorithms for finding low-rank approximations. J ACM (JACM) 51(6):1025–1041Google Scholar
  10. 10.
    Jolliffe IT (2002) Principal component analysis, vol 2. Wiley Online LibraryGoogle Scholar
  11. 11.
    Ke Q, Kanade T (2005) Robust l1 norm factorization in the presence of outliers and missing data by alternative convex programming. In: IEEE computer society conference on computer vision and pattern recognition (CVPR), vol 1. IEEE, pp 739–746Google Scholar
  12. 12.
    Kwak N (2008) Principal component analysis based on l1-norm maximization. IEEE Trans Pattern Anal Mach Intell 30(9):1672–1680Google Scholar
  13. 13.
    Mackey L (2009) Deflation methods for sparse pca. Adv Neural Inf Process Syst 21:1017–1024Google Scholar
  14. 14.
    Moghaddam B, Weiss Y, Avidan S (2006) Spectral bounds for sparse pca: exact and greedy algorithms. Adv Neural Inf Process Syst 18:915Google Scholar
  15. 15.
    Rousseeuw PJ, Leroy AM, Wiley J (1987) Robust regression and outlier detection, vol 3. Wiley Online LibraryGoogle Scholar
  16. 16.
    Shen H, Huang JZ (2008) Sparse principal component analysis via regularized low rank matrix approximation. J Multivar Anal 99(6):1015–1034Google Scholar
  17. 17.
    Wold S, Esbensen K, Geladi P (1987) Principal component analysis. Chemom Intell Lab Syst 2(1–3):37–52Google Scholar
  18. 18.
    Yang M, Zhang L, Yang J, Zhang D (2011) Robust sparse coding for face recognition. In: IEEE conference on computer vision and pattern recognition (CVPR). IEEE, pp 625–632Google Scholar
  19. 19.
    Zhou T, Tao D, Wu X (2011) Manifold elastic net: a unified framework for sparse dimension reduction. Data Min Knowl Dis 22(3):340–371Google Scholar
  20. 20.
    Zou H, Hastie T (2003) Regression shrinkage and selection via the elastic net, with applications to microarrays. In: Technical report. Department of Statistics, Stanford University. Available via DIALOG.
  21. 21.
    Zou H, Hastie T, Tibshirani R (2006) Sparse principal component analysis. J Comput Graph Stat 15(2):265–286Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.School of Electrical EngineeringUniversity of Electronic Science and Technology of ChinaChengduChina

Personalised recommendations