Skip to main content
SpringerLink
Log in

Manifold elastic net: a unified framework for sparse dimension reduction

  • Published:
Data Mining and Knowledge Discovery Aims and scope Submit manuscript

Abstract

It is difficult to find the optimal sparse solution of a manifold learning based dimensionality reduction algorithm. The lasso or the elastic net penalized manifold learning based dimensionality reduction is not directly a lasso penalized least square problem and thus the least angle regression (LARS) (Efron et al., Ann Stat 32(2):407–499, 2004), one of the most popular algorithms in sparse learning, cannot be applied. Therefore, most current approaches take indirect ways or have strict settings, which can be inconvenient for applications. In this paper, we proposed the manifold elastic net or MEN for short. MEN incorporates the merits of both the manifold learning based dimensionality reduction and the sparse learning based dimensionality reduction. By using a series of equivalent transformations, we show MEN is equivalent to the lasso penalized least square problem and thus LARS is adopted to obtain the optimal sparse solution of MEN. In particular, MEN has the following advantages for subsequent classification: (1) the local geometry of samples is well preserved for low dimensional data representation, (2) both the margin maximization and the classification error minimization are considered for sparse projection calculation, (3) the projection matrix of MEN improves the parsimony in computation, (4) the elastic net penalty reduces the over-fitting problem, and (5) the projection matrix of MEN can be interpreted psychologically and physiologically. Experimental evidence on face recognition over various popular datasets suggests that MEN is superior to top level dimensionality reduction algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Belhumeur PN, Hespanha JP, Kriegman DJ (1997) Eigenfaces vs. fisherfaces: recognition using class specific linear projection. IEEE Trans Pattern Anal Mach Intell 19(7): 711–720

    Article  Google Scholar 

  • Belkin M, Niyogi P (2001) Laplacian eigenmaps and spectral techniques for embedding and clustering. Adv Neural Inf Process Syst 14: 585–591

    Google Scholar 

  • Belkin M, Niyogi P, Sindhwani V (2006) Manifold regularization: a geometric framework for learning from labeled and unlabeled examples. J Mach Learn Res 7: 2399–2434

    MathSciNet  Google Scholar 

  • Bian W, Tao D (2008) Harmonic mean for subspace selection. In: IEEE ICPR, pp 1–4

  • Bishop CM, Williams CKI (1998) GTM: the generative topographic mapping. Neural Comput 10: 215–234

    Article  Google Scholar 

  • Cai D, He X, Han J (2007) Spectral regression for efficient regularized subspace learning. In: IEEE 11th international conference on computer vision, 2007. ICCV 2007, pp 1–8

  • Cai D, He X, Han J (2008) SRDA: an efficient algorithm for large-scale discriminant analysis. IEEE Trans Knowl Data Eng 20(1): 1–12

    Article  Google Scholar 

  • Candes E, Tao T (2005) The dantzig selector: statistical estimation when p is much larger than n. Ann Stat 35: 2392–2404

    Article  MathSciNet  Google Scholar 

  • Candes EJ, Wakin MB, Boyd SP (2008) Enhancing sparsity by reweighted L1 minimization. Special issue on sparsity. J Fourier Anal Appl 14(5): 877–905

    Article  MATH  MathSciNet  Google Scholar 

  • D’aspremont A, Ghaoui LE, Jordan MI, Lanckriet GRG (2007) A direct formulation for sparse PCA using semidefinite programming. SIAM Rev 49(3): 434–448

    Article  MATH  MathSciNet  Google Scholar 

  • Ding C, Li T (2007) Adaptive dimension reduction using discriminant analysis and k-means clustering. In: ICML ’07: Proceedings of the 24th international conference on machine learning. ACM, New York, NY, USA, pp 521–528

  • Ding CHQ, Li T, Jordan MI (2008) Convex and semi-nonnegative matrix factorizations. IEEE Trans Pattern Anal Mach Intell 32(1): 45–55

    Article  Google Scholar 

  • Donoho DL, Grimes C (2003) Hessian eigenmaps: locally linear embedding techniques for high-dimensional data. PNAS 100(10): 5591–5596

    Article  MATH  MathSciNet  Google Scholar 

  • Efron B, Hastie T, Johnstone I, Tibshirani R (2004) Least angle regression. Ann Stat 32(2): 407–499

    Article  MATH  MathSciNet  Google Scholar 

  • Fan J, Li R (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 96: 1348–1360

    Article  MATH  MathSciNet  Google Scholar 

  • Fan J, Lv J (2008) Sure independence screening for ultrahigh dimensional feature space. J R Stat Soc Ser B 70(5): 849–911

    Article  MathSciNet  Google Scholar 

  • Fisher RA (1936) The use of multiple measurements in taxonomic problems. Ann Eugen 7(2): 179–188

    Google Scholar 

  • Fyfe C (2007) Two topographic maps for data visualisation. Data Min Knowl Discov 14(2): 207–224

    Article  MathSciNet  Google Scholar 

  • Golub GH, Van Loan CF (1996) Matrix computations. 3. Johns Hopkins University Press, Baltimore, MD

    MATH  Google Scholar 

  • Graham DB, Allinson NM (1936) Characterizing virtual eigensignatures for general purpose face recognition. Face recognition: from theory to applications, NATO ASI series F. Comput Syst Sci 163: 446–456

    Google Scholar 

  • Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning: data mining, inference, and prediction, 2nd edn. Springer Series in Statistics, Springer, 2nd edn. corr. 3rd printing edn

  • He X, Niyogi P (2004) Locality preserving projections. In: Advances in neural information processing systems. MIT Press, Cambridge, MA

  • He X, Cai D, Yan S, Zhang HJ (2005a) Neighborhood preserving embedding. In: Proceedings of IEEE international conference on computer vision, vol. 2. IEEE Computer Society, Washington, DC, USA, pp 1208–1213

  • He X, Yan S, Hu Y, Niyogi P (2005b) Face recognition using laplacianfaces. IEEE Trans Pattern Anal Mach Intell 27(3): 328–340

    Article  Google Scholar 

  • Hotelling H (1936) Analysis of a complex of statistical variables into principal components. J Educ Phychol 24: 417–441

    Article  Google Scholar 

  • Huang H, Ding C (2008) Robust tensor factorization using r1 norm. In: IEEE conference on computer vision and pattern recognition, 2008. CVPR 2008, pp 1–8

  • James GM, Radchenko P, Lv J (2009) Dasso: connections between the dantzig selector and lasso. J R Stat Soc Ser B 71(1): 127–142

    Article  MATH  MathSciNet  Google Scholar 

  • Kriegel HP, Borgwardt KM, Kröger P, Pryakhin A, Schubert M, Zimek A (2007) Future trends in data mining. Data Min Knowl Discov 15(1): 87–97

    Article  MathSciNet  Google Scholar 

  • Li T, Zhu S, Ogihara M (2008a) Text categorization via generalized discriminant analysis. Inf Process Manage 44(5): 1684–1697

    Article  Google Scholar 

  • Li X, Lin S, Yan S, Xu D (2008b) Discriminant locally linear embedding with high-order tensor data. IEEE Trans Syst Man Cybern B 38(2): 342–352

    Article  Google Scholar 

  • Liu W, Tao D, Liu J (2008) Transductive component analysis. In: ICDM ’08: Proceedings of the 2008 eighth IEEE international conference on data mining. IEEE Computer Society, Washington, DC, USA, pp 433–442

  • Lv J, Fan Y (2009) A unified approach to model selection and sparse recovery using regularized least squares. Ann Stat 37: 3498

    Article  MATH  MathSciNet  Google Scholar 

  • Park MY, Hastie T (2006) L1 regularization path algorithm for generalized linear models. Tech. rep., Department of Statistics, Stanford University

  • Phillips PJ, Moon H, Rizvi SA, Rauss PJ (2000) The feret evaluation methodology for face-recognition algorithms. IEEE Trans Pattern Anal Mach Intell 22(10): 1090–1104

    Article  Google Scholar 

  • Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500): 2323–2326

    Article  Google Scholar 

  • Shakhnarovich G, Moghaddam B (2004) Face recognition in subspaces. In: Li SZ, Jain AK (eds) Handbook of face recognition. Springer-Verlag

  • Sun L, Ji S, Ye J (2008) A least squares formulation for canonical correlation analysis. In: ICML ’08: Proceedings of the 25th international conference on machine learning. ACM, New York, NY, USA, pp 1024–1031

  • Tao D, Li X, Wu X, Hu W, Maybank SJ (2007a) Supervised tensor learning. Knowl Inf Syst 13(1): 1–42

    Article  Google Scholar 

  • Tao D, Li X, Wu X, Maybank SJ (2007b) General tensor discriminant analysis and gabor features for gait recognition. IEEE Trans Pattern Anal Mach Intell 29(10): 1700–1715

    Article  Google Scholar 

  • Tao D, Li X, Wu X, Maybank SJ (2009) Geometric mean for subspace selection. IEEE Trans Pattern Anal Mach Intell 31(2): 260–274

    Article  Google Scholar 

  • Tenenbaum JB (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500): 2319–2323

    Article  Google Scholar 

  • Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B 58: 267–288

    MATH  MathSciNet  Google Scholar 

  • Turk MA, Pentland AP (1991) Face recognition using eigenfaces. In: IEEE computer society conference on computer vision and pattern recognition, 1991. Proceedings CVPR ’91, pp 586–591

  • Wang F, Chen S, Zhang C, Li T (2008) Semi-supervised metric learning by maximizing constraint margin. In: CIKM ’08: Proceeding of the 17th ACM conference on information and knowledge management. ACM, New York, NY, USA, pp 1457–1458

  • Yan S, Xu D, Zhang B, Zhang HJ, Yang Q, Lin S (2007) Graph embedding and extensions: a general framework for dimensionality reduction. IEEE Trans Pattern Anal Mach Intell 29(1): 40–51

    Article  Google Scholar 

  • Ye J (2007) Least squares linear discriminant analysis. In: ICML ’07: Proceedings of the 24th international conference on machine learning. ACM, New York, NY, USA, pp 1087–1093

  • Zass R, Shashua A (2007) Nonnegative sparse PCA. In: In neural information processing systems. pp 1561–1568

  • Zhang Z, Zha H (2005) Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM J Sci Comput 26(1): 313–338

    Article  MathSciNet  Google Scholar 

  • Zhang T, Tao D, Yang J (2008) Discriminative locality alignment. In: ECCV ’08: Proceedings of the 10th European conference on computer vision. Springer-Verlag, Berlin, Heidelberg, pp 725–738

  • Zhang T, Tao D, Li X, Yang J (2009) Patch alignment for dimensionality reduction. IEEE Trans Knowl Data Eng 21(9): 1299–1313

    Article  Google Scholar 

  • Zou H (2006) The adaptive Lasso and its Oracle properties. J Am Stat Assoc 101(476): 1418–1429

    Article  MATH  Google Scholar 

  • Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc B 67: 301–320

    Article  MATH  MathSciNet  Google Scholar 

  • Zou H, Hastie T, Tibshirani R (2006) Sparse principal component analysis. J Comput Graph Stat 15(2): 262–286

    MathSciNet  Google Scholar 

  • Zou H, Zhang HH (2009) On the adaptive elastic-net with a diverging number of parameters. Ann Stat 37(4): 1733–1751

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer Engineering, Nanyang Technological University, Singapore, 639798, Singapore

    Tianyi Zhou & Dacheng Tao

  2. Department of Computer Science, University of Vermont, 33 Colchester Avenue, Burlington, VT, 05405, USA

    Xindong Wu

Authors
  1. Tianyi Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dacheng Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xindong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dacheng Tao.

Additional information

Responsible editors: Tao Li, Chris Ding and Fei Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, T., Tao, D. & Wu, X. Manifold elastic net: a unified framework for sparse dimension reduction. Data Min Knowl Disc 22, 340–371 (2011). https://doi.org/10.1007/s10618-010-0182-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10618-010-0182-x

Keywords

Search

Navigation