Skip to main content

2016 | OriginalPaper | Buchkapitel

A Simple Review of Sparse Principal Components Analysis

verfasst von : Chun-Mei Feng, Ying-Lian Gao, Jin-Xing Liu, Chun-Hou Zheng, Sheng-Jun Li, Dong Wang

Erschienen in: Intelligent Computing Theories and Application

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Principal Component Analysis (PCA) is a common tool for dimensionality reduction and feature extraction, which has been applied in many fields, such as biology, medicine, machine learning and bioinformatics. But PCA has two obvious drawbacks: each principal component is line combination and loadings are non-zero which is hard to interpret. Sparse Principal Component Analysis (SPCA) was proposed to overcome these two disadvantages of PCA under the circumstances. This review paper will mainly focus on the research about SPCA, where the basic models of PCA and SPCA, various algorithms and extensions of SPCA are summarized. According to the difference of objective function and the constraint conditions, SPCA can be divided into three groups as it shown in Fig. 1. We also make a comparison among the different kind of sparse penalties. Besides, brief statements and other different classifications are summarized at last.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Liu, J.-X., Xu, Y., Gao, Y.-L., Zheng, C.-H., Wang, D., Zhu, Q.: A Class-Information-based Sparse Component Analysis Method to Identify Differentially Expressed Genes on RNA-Seq Data. 13(2), 392–398 (2015) Liu, J.-X., Xu, Y., Gao, Y.-L., Zheng, C.-H., Wang, D., Zhu, Q.: A Class-Information-based Sparse Component Analysis Method to Identify Differentially Expressed Genes on RNA-Seq Data. 13(2), 392–398 (2015)
2.
Zurück zum Zitat Richtárik, P., Takáč, M., Ahipaşaoğlu, S.D.: Alternating maximization: unifying framework for 8 sparse PCA formulations and efficient parallel codes. arXiv preprint arXiv:1212.4137 (2012) Richtárik, P., Takáč, M., Ahipaşaoğlu, S.D.: Alternating maximization: unifying framework for 8 sparse PCA formulations and efficient parallel codes. arXiv preprint arXiv:​1212.​4137 (2012)
3.
Zurück zum Zitat Hotelling, H.: Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24(6), 417 (1933)CrossRefMATH Hotelling, H.: Analysis of a complex of statistical variables into principal components. J. Educ. Psychol. 24(6), 417 (1933)CrossRefMATH
4.
Zurück zum Zitat Jolliffe, I.T.: Rotation of principal components: choice of normalization constraints. J. Appl. Stat. 22(1), 29–35 (1995)MathSciNetCrossRef Jolliffe, I.T.: Rotation of principal components: choice of normalization constraints. J. Appl. Stat. 22(1), 29–35 (1995)MathSciNetCrossRef
6.
Zurück zum Zitat Moghaddam, B., Weiss, Y., Avidan, S.: Spectral bounds for sparse PCA: Exact and greedy algorithms. In: Advances in neural information processing systems, pp. 915–922 (2005) Moghaddam, B., Weiss, Y., Avidan, S.: Spectral bounds for sparse PCA: Exact and greedy algorithms. In: Advances in neural information processing systems, pp. 915–922 (2005)
7.
Zurück zum Zitat d’Aspremont, A., Bach, F.R., Ghaoui, L.E.: Full regularization path for sparse principal component analysis. In: Proceedings of the 24th international conference on Machine learning, pp. 177–184. ACM (2007) d’Aspremont, A., Bach, F.R., Ghaoui, L.E.: Full regularization path for sparse principal component analysis. In: Proceedings of the 24th international conference on Machine learning, pp. 177–184. ACM (2007)
9.
Zurück zum Zitat Mackey, L.W.: Deflation methods for sparse PCA. In: Advances in Neural Information Processing Systems, pp. 1017–1024 (2009) Mackey, L.W.: Deflation methods for sparse PCA. In: Advances in Neural Information Processing Systems, pp. 1017–1024 (2009)
11.
Zurück zum Zitat Kuleshov, V.: Fast algorithms for sparse principal component analysis based on Rayleigh quotient iteration. In: Proceedings of the 30th International Conference on Machine Learning (ICML-13), pp. 1418–1425 (2013) Kuleshov, V.: Fast algorithms for sparse principal component analysis based on Rayleigh quotient iteration. In: Proceedings of the 30th International Conference on Machine Learning (ICML-13), pp. 1418–1425 (2013)
12.
Zurück zum Zitat Journée, M., Nesterov, Y., Richtárik, P., Sepulchre, R.: Generalized power method for sparse principal component analysis. J. Mach. Learn. Res. 11, 517–553 (2010)MathSciNetMATH Journée, M., Nesterov, Y., Richtárik, P., Sepulchre, R.: Generalized power method for sparse principal component analysis. J. Mach. Learn. Res. 11, 517–553 (2010)MathSciNetMATH
13.
Zurück zum Zitat Yuan, X.-T., Zhang, T.: Truncated power method for sparse eigenvalue problems. J. Mach. Learn. Res. 14(1), 899–925 (2013)MathSciNetMATH Yuan, X.-T., Zhang, T.: Truncated power method for sparse eigenvalue problems. J. Mach. Learn. Res. 14(1), 899–925 (2013)MathSciNetMATH
14.
Zurück zum Zitat Zhao, Q., Meng, D., Xu, Z.: A recursive divide-and-conquer approach for sparse principal component analysis. arXiv preprint, arXiv:1211.7219 (2012) Zhao, Q., Meng, D., Xu, Z.: A recursive divide-and-conquer approach for sparse principal component analysis. arXiv preprint, arXiv:​1211.​7219 (2012)
15.
Zurück zum Zitat Zass, R., Shashua, A.: Nonnegative sparse PCA. In: Advances in Neural Information Processing Systems, pp. 1561–1568 (2006) Zass, R., Shashua, A.: Nonnegative sparse PCA. In: Advances in Neural Information Processing Systems, pp. 1561–1568 (2006)
16.
17.
Zurück zum Zitat Jolliffe, I.T., Trendafilov, N.T., Uddin, M.: A modified principal component technique based on the LASSO. J. Comput. Graph. Stat. 12(3), 531–547 (2003)MathSciNetCrossRef Jolliffe, I.T., Trendafilov, N.T., Uddin, M.: A modified principal component technique based on the LASSO. J. Comput. Graph. Stat. 12(3), 531–547 (2003)MathSciNetCrossRef
18.
Zurück zum Zitat Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. Ser. B (Methodological) 58, 267–288 (1996)MathSciNetMATH Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Roy. Stat. Soc. Ser. B (Methodological) 58, 267–288 (1996)MathSciNetMATH
19.
Zurück zum Zitat d’Aspremont, A., El Ghaoui, L., Jordan, M.I., Lanckriet, G.R.: A direct formulation for sparse PCA using semidefinite programming. SIAM Rev. 49(3), 434–448 (2007)MathSciNetCrossRefMATH d’Aspremont, A., El Ghaoui, L., Jordan, M.I., Lanckriet, G.R.: A direct formulation for sparse PCA using semidefinite programming. SIAM Rev. 49(3), 434–448 (2007)MathSciNetCrossRefMATH
20.
Zurück zum Zitat d’Aspremont, A., Bach, F., Ghaoui, L.E.: Optimal solutions for sparse principal component analysis. J. Mach. Learn. Res. 9, 1269–1294 (2008)MathSciNetMATH d’Aspremont, A., Bach, F., Ghaoui, L.E.: Optimal solutions for sparse principal component analysis. J. Mach. Learn. Res. 9, 1269–1294 (2008)MathSciNetMATH
21.
Zurück zum Zitat Lu, Z., Zhang, Y.: An augmented Lagrangian approach for sparse principal component analysis. Math. Program. 135(1–2), 149–193 (2012)MathSciNetCrossRefMATH Lu, Z., Zhang, Y.: An augmented Lagrangian approach for sparse principal component analysis. Math. Program. 135(1–2), 149–193 (2012)MathSciNetCrossRefMATH
22.
Zurück zum Zitat Guo, J., James, G., Levina, E., Michailidis, G., Zhu, J.: Principal component analysis with sparse fused loadings. J. Comput. Graph. Stat. 19(4), 930–946 (2012)MathSciNetCrossRef Guo, J., James, G., Levina, E., Michailidis, G., Zhu, J.: Principal component analysis with sparse fused loadings. J. Comput. Graph. Stat. 19(4), 930–946 (2012)MathSciNetCrossRef
23.
24.
Zurück zum Zitat Hein, M., Bühler, T.: An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. In: Advances in Neural Information Processing Systems, pp. 847–855 (2010) Hein, M., Bühler, T.: An inverse power method for nonlinear eigenproblems with applications in 1-spectral clustering and sparse PCA. In: Advances in Neural Information Processing Systems, pp. 847–855 (2010)
25.
Zurück zum Zitat Zou, H., Hastie, T., Tibshirani, R.: Sparse principal component analysis. J. Comput. Graph. Stat. 15(2), 265–286 (2006)MathSciNetCrossRef Zou, H., Hastie, T., Tibshirani, R.: Sparse principal component analysis. J. Comput. Graph. Stat. 15(2), 265–286 (2006)MathSciNetCrossRef
26.
Zurück zum Zitat Leng, C., Wang, H.: On general adaptive sparse principal component analysis. J. Comput. Graph. Stat. 18, 201–215 (2012)MathSciNetCrossRef Leng, C., Wang, H.: On general adaptive sparse principal component analysis. J. Comput. Graph. Stat. 18, 201–215 (2012)MathSciNetCrossRef
27.
Zurück zum Zitat Xiaoshuang, S., Zhihui, L., Zhenhua, G., Minghua, W., Cairong, Z., Heng, K.: Sparse Principal Component Analysis via Joint L 2, 1-Norm Penalty. In: Cranefield, S., Nayak, A. (eds.) AI 2013. LNCS, vol. 8272, pp. 148–159. Springer, Heidelberg (2013)CrossRef Xiaoshuang, S., Zhihui, L., Zhenhua, G., Minghua, W., Cairong, Z., Heng, K.: Sparse Principal Component Analysis via Joint L 2, 1-Norm Penalty. In: Cranefield, S., Nayak, A. (eds.) AI 2013. LNCS, vol. 8272, pp. 148–159. Springer, Heidelberg (2013)CrossRef
28.
Zurück zum Zitat Witten, D.M., Tibshirani, R., Hastie, T.: A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10, 515–534 (2009). kxp008CrossRef Witten, D.M., Tibshirani, R., Hastie, T.: A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. Biostatistics 10, 515–534 (2009). kxp008CrossRef
29.
Zurück zum Zitat Shen, H., Huang, J.Z.: Sparse principal component analysis via regularized low rank matrix approximation. J. Multivar. Anal. 99(6), 1015–1034 (2008)MathSciNetCrossRefMATH Shen, H., Huang, J.Z.: Sparse principal component analysis via regularized low rank matrix approximation. J. Multivar. Anal. 99(6), 1015–1034 (2008)MathSciNetCrossRefMATH
30.
Zurück zum Zitat Papailiopoulos, D.S., Dimakis, A.G., Korokythakis, S.: Sparse PCA through low-rank approximations. arXiv preprint, arXiv:1303.0551 (2013) Papailiopoulos, D.S., Dimakis, A.G., Korokythakis, S.: Sparse PCA through low-rank approximations. arXiv preprint, arXiv:​1303.​0551 (2013)
31.
Zurück zum Zitat Lee, D., Lee, W., Lee, Y., Pawitan, Y.: Super-sparse principal component analyses for high-throughput genomic data. BMC Bioinformatics 11(1), 1 (2010)CrossRef Lee, D., Lee, W., Lee, Y., Pawitan, Y.: Super-sparse principal component analyses for high-throughput genomic data. BMC Bioinformatics 11(1), 1 (2010)CrossRef
32.
Zurück zum Zitat Meng, D., Zhao, Q., Xu, Z.: Improve robustness of sparse PCA by L 1-norm maximization. Pattern Recogn. 45(1), 487–497 (2012)CrossRefMATH Meng, D., Zhao, Q., Xu, Z.: Improve robustness of sparse PCA by L 1-norm maximization. Pattern Recogn. 45(1), 487–497 (2012)CrossRefMATH
33.
Zurück zum Zitat Jenatton, R., Obozinski, G., Bach, F.: Structured sparse principal component analysis. arXiv preprint arXiv:0909.1440 (2009) Jenatton, R., Obozinski, G., Bach, F.: Structured sparse principal component analysis. arXiv preprint arXiv:​0909.​1440 (2009)
34.
Zurück zum Zitat Tipping, M.E., Nh, C.C.: Sparse kernel principal component analysis (2001) Tipping, M.E., Nh, C.C.: Sparse kernel principal component analysis (2001)
35.
Zurück zum Zitat Lounici, K.: Sparse principal component analysis with missing observations. In: Houdré, C., Mason, D.M., Rosiński, J., Wellner, J.A. (eds.) High dimensional probability VI, vol. 66, pp. 327–356. Springer, Heidelberg (2013)CrossRef Lounici, K.: Sparse principal component analysis with missing observations. In: Houdré, C., Mason, D.M., Rosiński, J., Wellner, J.A. (eds.) High dimensional probability VI, vol. 66, pp. 327–356. Springer, Heidelberg (2013)CrossRef
36.
Zurück zum Zitat Asteris, M., Papailiopoulos, D.S., Karystinos, G.N.: Sparse principal component of a rank-deficient matrix. In: IEEE International Symposium on 2011 Information Theory Proceedings (ISIT), pp. 673–677. IEEE (2011) Asteris, M., Papailiopoulos, D.S., Karystinos, G.N.: Sparse principal component of a rank-deficient matrix. In: IEEE International Symposium on 2011 Information Theory Proceedings (ISIT), pp. 673–677. IEEE (2011)
37.
Zurück zum Zitat Liu, W., Zhang, H., Tao, D., Wang, Y., Lu, K.: Large-scale paralleled sparse principal component analysis. Multimedia Tools Appl. 1–13 (2014) Liu, W., Zhang, H., Tao, D., Wang, Y., Lu, K.: Large-scale paralleled sparse principal component analysis. Multimedia Tools Appl. 1–13 (2014)
38.
Zurück zum Zitat Xiao, C.: Two-dimensional sparse principal component analysis for face recognition. In: 2010 2nd International Conference on Future Computer and Communication (ICFCC), pp. V2–561-V562-565. IEEE (2010) Xiao, C.: Two-dimensional sparse principal component analysis for face recognition. In: 2010 2nd International Conference on Future Computer and Communication (ICFCC), pp. V2–561-V562-565. IEEE (2010)
39.
Zurück zum Zitat Lai, Z., Xu, Y., Chen, Q., Yang, J., Zhang, D.: Multilinear sparse principal component analysis. IEEE Trans. Neural Netw. Learn. Syst. 25(10), 1942–1950 (2014)CrossRef Lai, Z., Xu, Y., Chen, Q., Yang, J., Zhang, D.: Multilinear sparse principal component analysis. IEEE Trans. Neural Netw. Learn. Syst. 25(10), 1942–1950 (2014)CrossRef
40.
Zurück zum Zitat Sharp, K., Rattray, M.: Dense message passing for sparse principal component analysis. In: International Conference on Artificial Intelligence and Statistics, pp. 725–732 (2010) Sharp, K., Rattray, M.: Dense message passing for sparse principal component analysis. In: International Conference on Artificial Intelligence and Statistics, pp. 725–732 (2010)
41.
Zurück zum Zitat Wang, S.-J., Sun, M.-F., Chen, Y.-H., Pang, E.-P., Zhou, C.-G.: STPCA: sparse tensor principal component analysis for feature extraction. In: 21st International Conference on 2012 Pattern Recognition (ICPR), pp. 2278–2281. IEEE (2012) Wang, S.-J., Sun, M.-F., Chen, Y.-H., Pang, E.-P., Zhou, C.-G.: STPCA: sparse tensor principal component analysis for feature extraction. In: 21st International Conference on 2012 Pattern Recognition (ICPR), pp. 2278–2281. IEEE (2012)
42.
Zurück zum Zitat Jiang, R., Fei, H., Huan, J.: Anomaly localization for network data streams with graph joint sparse PCA. In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 886–894. ACM (2011) Jiang, R., Fei, H., Huan, J.: Anomaly localization for network data streams with graph joint sparse PCA. In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 886–894. ACM (2011)
43.
Zurück zum Zitat Grbovic, M., Li, W., Xu, P., Usadi, A.K., Song, L., Vucetic, S.: Decentralized fault detection and diagnosis via sparse PCA based decomposition and Maximum Entropy decision fusion. J. Process Control 22(4), 738–750 (2012)CrossRef Grbovic, M., Li, W., Xu, P., Usadi, A.K., Song, L., Vucetic, S.: Decentralized fault detection and diagnosis via sparse PCA based decomposition and Maximum Entropy decision fusion. J. Process Control 22(4), 738–750 (2012)CrossRef
44.
Zurück zum Zitat Johnstone, I.M., Lu, A.Y.: On consistency and sparsity for principal components analysis in high dimensions. J. Am. Stat. Assoc. 104 (2012) Johnstone, I.M., Lu, A.Y.: On consistency and sparsity for principal components analysis in high dimensions. J. Am. Stat. Assoc. 104 (2012)
45.
Zurück zum Zitat Shen, D., Shen, H., Marron, J.S.: Consistency of sparse PCA in high dimension, low sample size contexts. J. Multivar. Anal. 115, 317–333 (2013)MathSciNetCrossRefMATH Shen, D., Shen, H., Marron, J.S.: Consistency of sparse PCA in high dimension, low sample size contexts. J. Multivar. Anal. 115, 317–333 (2013)MathSciNetCrossRefMATH
46.
47.
48.
Zurück zum Zitat Zhang, Y., d’Aspremont, A., El Ghaoui, L.: Sparse PCA: Convex relaxations, algorithms and applications. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite, Conic and Polynomial Optimization, vol. 166, pp. 915–940. Springer, Heidelberg (2012)CrossRef Zhang, Y., d’Aspremont, A., El Ghaoui, L.: Sparse PCA: Convex relaxations, algorithms and applications. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on Semidefinite, Conic and Polynomial Optimization, vol. 166, pp. 915–940. Springer, Heidelberg (2012)CrossRef
49.
Zurück zum Zitat Wang, Z., Lu, H., Liu, H.: Tighten after relax: Minimax-optimal sparse PCA in polynomial time. Adv. Neural Inf. Process. Syst. 2014, 3383–3391 (2014) Wang, Z., Lu, H., Liu, H.: Tighten after relax: Minimax-optimal sparse PCA in polynomial time. Adv. Neural Inf. Process. Syst. 2014, 3383–3391 (2014)
51.
Zurück zum Zitat Johnstone, I.M., Lu, A.Y.: Sparse principal components analysis. Unpublished manuscript 7 (2004) Johnstone, I.M., Lu, A.Y.: Sparse principal components analysis. Unpublished manuscript 7 (2004)
52.
Zurück zum Zitat Ulfarsson, M.O., Solo, V.: Sparse variable PCA using geodesic steepest descent. IEEE Trans. Sig. Process. 56(12), 5823–5832 (2008)MathSciNetCrossRef Ulfarsson, M.O., Solo, V.: Sparse variable PCA using geodesic steepest descent. IEEE Trans. Sig. Process. 56(12), 5823–5832 (2008)MathSciNetCrossRef
Metadaten
Titel
A Simple Review of Sparse Principal Components Analysis
verfasst von
Chun-Mei Feng
Ying-Lian Gao
Jin-Xing Liu
Chun-Hou Zheng
Sheng-Jun Li
Dong Wang
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-42294-7_33