Non-Negative Sparse Principal Component Analysis

Authors

  • Thanh D. X. Duong Ton Duc Thang University, Vietnam

Corressponding author's email:

thanhddx@tut.edu.vn

Keywords:

principal component analysis, semi-definite relaxation, semi-definite programming, 1 -minimization, iterative reweighting, bisection algorithm

Abstract

With applications throughout science and engineering, sparse principal component analysis considers the problem of maximizing the variance explained by a particular linear combination of the input variables where the number ofnonzero coefficients is constrained. In this paper, we consider nonnegative sparse principal component analysis in which the coeficients in the combination are required to be non-negative. We improve our recent results by applying an appropriate re-weighted algorithms. Numerical results illustrate the improvement.

Downloads: 0

Download data is not yet available.

References

Alizadeh, F.: Interior point methods in semi-definite programming with applications to combinatorial optimization. SIAM J. Optim. 5 (1995) 13-51

Badea, L., Tilivea, D.: Sparse factorizations of gene expression guided by binding data. In Pacific Symposium on Biocomputing (2005)

Boyd, S., Vandenberghe,L.: Convex optimization. Cambridge University Press. Cambridge UK (2004)

Cadima,J., Jolliffe, I.T.: Loadings and correlations in the interpretation of principal components. J. Appl. Statist. 22 (1995) 203-214

Candes, E.J., Wakin, M.B., Boyd, S.: Enhancing sparsity by re-weighted I l minimization. (preprint)

D'Aspremont, A. , El Ghaoui, L. , Jordan, M.I., Lanckriet, G.R.G.: A direct formulation for sparse PCA using semi-definite programming. SIAM Rev. 49 (2007) 434-448

Duong, V.: Dynamic models for airborne air traffic management capability: State-ofthe-art analysis. (Internal report) Bretigny, France: Eurocontrol Experimental Centre (1996)

Duong, D.X.T, Duong, V.: Non-Negative Sparse Principal Component Analysis for Multidimensional Constrained Optimization, The Tenth Pacific Rim International Conference on Artificial Intelligence (PRICAI 08), LNAI 5351, Springer-Verlag Berlin Heidelberg, (2008) 103-114.

Duong, D.X.T, Duong, V.: Exact and Direct Approach for Sparse Principal Component Analysis, Vietnam Journal of Science and Technology, 46 (2008) 7-22

Duong, D.X.T, Duong, V.: Principal Component Analysis with Weighted Sparsity Constraint, (2009) (to appear in Applied Mathematics & Information Sciences)

l l . Fazel, M., Hindi, H. , Boyd, S.: A rank minimization heuristic with application to minimum order system approximation. Proceedings of the American Control Conference, Arlington. VA. Vol. 6 (2001) 4734-4739

Hotelling, H.: Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24 (1933) 417-441

Jagannathan, R. , Ma, T.: Risk reduction in large portfolios: Why imposing the wrong constraints helps. Journal of Finance 58 (2003) 1651-1684

Jeffers, J.: Two case studies in the application of principal components. Appl. Statist. 16 (1967) 225-236

Jolliffe, I.T.: Rotation of principal components: Choice of normalization constraints. J. Appl. Statist. 22 (1995) 29-35.

Jolliffe, I.T., Trendafilov, N.T., Uddin, M.: A modified principal component technique based on the LASSO. J.Comput. Graphical Statist. 12 (2003) 531-547

Jolliffe, I.T.: Principal component analysis. Springer Verlag, New York (2002)

Lemarechal, C. , Oustry, F.: Semi-definite relaxations and lagrangian duality with application to combinatorial optimization. Rapport de recherche 3710, INRIA, France (1999)

Lovasz, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. optim. 1 (1991) 166-190

Moghaddam, B., Weiss, Y., Avidan, S.: Spectral Bounds for Sparse PCA: Exact & Greedy Algorithms. Advances in Neural Information Processing Systems 18 (2006) 915-922. MIT Press.

Nesterov, Y.: Smoothing technique and its application in semi-definite optimization. Math. Program. 110 (2007) 245-259

Pearson, K.: On lines and planes of closest fit to systems of points in space. Phil. Mag. 2 (1901), 559-572

Sturm, J.: Using SEDUMI 1.0x, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. I l (1999) 625-653

Toh, K.C., Todd, M.J., Tutuncu, R.H.: SDPT3 - a MA TLAB software package for semi-definite programming. Optim. Methods Softw. 11 (1999) 545-581

Vines, S.: Simple principal components. Appl. Statist. 49 (2000) 441-451

Zass, R., Shashua, A.: Non-negative Sparse PCA. Advances In Neural Information Processing Systems 19 (2007) 1561-1568

Zhang, Z., Zha, H., Simon, H.: Low-rank approximations with sparse factors I: Basic algorithms and error analysis. SIAM J. Matrix Anal. Appl. 23 (2002) 706-727

Zhang, Z., Zha, H., Simon, H.: Low-rank approximations with sparse factors Il: Penalized methods with discrete Newton-like iterations. SIAM J. Matrix Anal. Appl. 25 (2004) 901-920

Zou, H. , Hastie, T., Tibshirani, R.: Sparse principal component analysis. J. Comput. Graphical Statist. 15 (2006) 265-286

Downloads

Published

28-12-2009

How to Cite

[1]
Thanh D. X. Duong, “Non-Negative Sparse Principal Component Analysis”, JTE, vol. 4, no. 3, pp. 31–38, Dec. 2009.

Issue

Section

Research Article

Categories