Abstract
This chapter begins with the motivation of sparse PCA–to improve the physical interpretation of the loadings. Second, we introduce the issues involved in sparse PCA problem that are distinct from PCA problem. Third, we briefly review some sparse PCA algorithms in the literature, and comment their limitations as well as problems unresolved. Forth, we introduce one of the state-of-the-art algorithms, SPCArt Hu et al. (IEEE Trans. Neural Networks Learn. Syst. 27(4):875–890, 2016), including its motivating idea, formulation, optimization solution, and performance analysis. Along with the introduction, we describe how SPCArt addresses the unresolved problems. Fifth, based on the Eckart-Young Theorem, we provide a unified view to a series of sparse PCA algorithms including SPCArt. Finally, we make a concluding remark.
This is a preview of subscription content, log in via an institution.
Notes
- 1.
Theorem 13 is specific to SPCArt, which concerns the important explained variance. The other results are applicable to more general situations: Propositions 6–11 are applicable to any orthonormal Z, Theorem 12 is applicable to any matrix X. To obtain results specific to SPCArt, some assumptions of the data distribution are needed.
- 2.
[21] did implement this version for rSVD, but using a heuristic approach.
References
Amini, A., Wainwright, M.: High-dimensional analysis of semidefinite relaxations for sparse principal components. Ann. Stat. 37(5B), 2877–2921 (2009)
Cadima, J., Jolliffe, I.: Loading and correlations in the interpretation of principle components. J. Appl. Stat. 22(2), 203–214 (1995)
d’Aspremont, A., Bach, F., Ghaoui, L.: Optimal solutions for sparse principal component analysis. J. Mach. Learn. Res. 9, 1269–1294 (2008)
d’Aspremont, A., El Ghaoui, L., Jordan, M., Lanckriet, G.: A direct formulation for sparse pca using semidefinite programming. SIAM Rev. 49(3), 434–448 (2007)
Donoho, D.L.: For most large underdetermined systems of linear equations the minimal \(\ell \)1-norm solution is also the sparsest solution. Commun. Pure Appl. Math. 59(6), 797–829 (2006)
Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrika 1(3), 211–218 (1936)
Fan, K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142(3), 305–310 (1961)
Golub, G., Van Loan, C.: Matrix Computations, vol. 3. Johns Hopkins University Press, Baltimore (1996)
Hu, Z., Pan, G., Wang, Y., Wu, Z.: Sparse principal component analysis via rotation and truncation. IEEE Trans. Neural Networks Learn. Syst. 27(4), 875–890 (2016)
Jolliffe, I.: Principal Component Analysis. Springer, Berlin (2002)
Jolliffe, I., Trendafilov, N., Uddin, M.: A modified principal component technique based on the lasso. J. Comput. Graphical Stat. 12(3), 531–547 (2003)
Jolliffe, I.T.: Rotation of ill-defined principal components. Appl. Stat. pp. 139–147 (1989)
Journée, M., Nesterov, Y., Richtárik, P., Sepulchre, R.: Generalized power method for sparse principal component analysis. J. Mach. Learn. Res. 11, 517–553 (2010)
Lai, Z., Xu, Y., Chen, Q., Yang, J., Zhang, D.: Multilinear sparse principal component analysis. IEEE Trans. Neural Networks Learn. Syst. 25(10), 1942–1950 (2014)
Lu, Z., Zhang, Y.: An augmented Lagrangian approach for sparse principal component analysis. Math. Program. 135(1–2), 149–193 (2012)
Ma, Z.: Sparse principal component analysis and iterative thresholding. Ann. Stat. 41(2), 772–801 (2013)
Mackey, L.: Deflation methods for sparse PCA. Adv. Neural Inf. Process. Syst. 21, 1017–1024 (2009)
Moghaddam, B., Weiss, Y., Avidan, S.: Generalized spectral bounds for sparse LDA. In: Proceedings of the 23rd International Conference on Machine Learning, pp. 641-648. ACM, New York (2006)
Moghaddam, B., Weiss, Y., Avidan, S.: Spectral bounds for sparse PCA: exact and greedy algorithms. Adv. Neural Inf. Process. Syst. 18, 915 (2006)
Paul, D., Johnstone, I.M.: Augmented sparse principal component analysis for high dimensional data. arXiv preprint arXiv:1202.1242, (2012)
Shen, H., Huang, J.: Sparse principal component analysis via regularized low rank matrix approximation. J. Multivar. Anal. 99(6), 1015–1034 (2008)
Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Royal Stat. Soc. Series B (Methodol.), pp. 267–288 (1996)
Tseng, P.: Convergence of a block coordinate descent method for nondifferentiable minimization. J. Optim. Theory Appl. 109(3), 475–494 (2001)
Witten, D., Tibshirani, R., Hastie, T.: A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10(3), 515 (2009)
Yuan, X., Zhang, T.: Truncated power method for sparse eigenvalue problems. J. Mach. Learn. Res. 14, 899–925 (2013)
Zhang, Y., d’Aspremont, A., Ghaoui, L.: Sparse pca: convex relaxations, algorithms and applications. In: Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 915–940 (2012)
Zhang, Y., Ghaoui, L.E.: Large-scale sparse principal component analysis with application to text data. In: Advances in Neural Information Processing Systems (2011)
Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. Royal Stat. Soc.: Series B (Stat. Methodol.) 67(2), 301–320 (2005)
Zou, H., Hastie, T., Tibshirani, R.: Sparse principal component analysis. J. Comput. Graphical Stat. 15(2), 265–286 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Hu, Z., Pan, G., Wang, Y., Wu, Z. (2018). Sparse Principal Component Analysis via Rotation and Truncation. In: Naik, G. (eds) Advances in Principal Component Analysis. Springer, Singapore. https://doi.org/10.1007/978-981-10-6704-4_1
Download citation
DOI: https://doi.org/10.1007/978-981-10-6704-4_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-6703-7
Online ISBN: 978-981-10-6704-4
eBook Packages: EngineeringEngineering (R0)