Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Sparse Principal Component Analysis via Rotation and Truncation Read chapter Read first chapter

2018 | OriginalPaper | Chapter

Robust PCAs and PCA Using Generalized Mean

Authors : Jiyong Oh, Nojun Kwak

Published in: Advances in Principal Component Analysis

Publisher: Springer Singapore

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter, a robust principal component analysis (PCA) is described, which can overcome the problem that PCA is prone to outliers included in training set. Different from the other alternatives which commonly replace \(L_{2}\)-norm by other distance measures, our method alleviates the negative effect of outliers using the characteristic of the generalized mean keeping the use of the Euclidean distance. The optimization problem based on the generalized mean is solved by a novel method. We also present a generalized sample mean, which is a generalization of the sample mean, to estimate a robust mean in the presence of outliers. The proposed method shows better or equivalent performance than the conventional PCAs in various problems such as face reconstruction, clustering, and object categorization.

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

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!

inform now
previous chapter Principal Component Analysis in the Presence of Missing Data
next chapter Principal Component Analysis Techniques for Visualization of Volumetric Data
Appendix
Available only for authorised users
Footnotes
Literature
1.
Jain, A., Duin, R., Jianchang, M.: Statistical pattern recognition: a review. IEEE Trans. Pattern Anal. Machine Intell. 22(1), 4–37 (2000)CrossRef
2.
Jolliffe, I.: Principal Component Analysis, 2nd edn. Springer, Berlin (2002)MATH
3.
Belhumeur, P., Hespanha, J., Kriegman, D.: Eigenfaces versus fisherfaces: recognition using class specific linear projection. IEEE Trans. Pattern Anal. Machine Intell. 19(7), 711–720 (1997)CrossRef
4.
Turk, M., Pentland, A.: Eigenfaces for recognition. J. Cognitive Neurosci. 3(1), 71–86 (1991)CrossRef
5.
Ross, D.A., Lim, J., Lin, R.S., Yang, M.H.: Incremental learning for robust visual tracking. Int. J. Comput. Vision 77(1–3), 125–141 (2008)CrossRef
6.
Ding, C., He, X.: \(K\)-means Clustering via Principal Component Analysis. In: Proceedings of the 21st International Conference on Machine Learning, ICML ’04 (2004)
7.
Yeung, K.Y., Ruzzo, W.L.: Principal component analysis for clustering gene expression data. Bioinformatics 17(9), 763–774 (2001)CrossRef
8.
de la Torre, F., Black, M.J.: A framework for robust subspace learning. Int. J. Comput. Vision 54(1–3), 117–142 (2003)CrossRefMATH
9.
Brooks, J., Dulá, J., Boone, E.: A pure \(L_{1}\)-norm principal component analysis. Comput. Statist. Data Anal. 61, 83–98 (2013)MathSciNetCrossRefMATH
10.
Ding, C., Zhou, D., He, X., Zha, H.: \(R_{1}\)-PCA: Rotational Invariant \(L_{1}\)-norm Principal Component Analysis for Robust Subspace Factorization. In: Proceedings of the 23rd International Conference on Machine Learning, ICML ’06, pp. 281–288 (2006)
11.
He, R., Hu, B., Yuan, X., Zheng, W.S.: Principal component analysis based on non-parametric maximum entropy. Neurocomputing 73(10–12), 1840–1852 (2010)CrossRef
12.
He, R., Hu, B.G., Zheng, W.S., Kong, X.W.: Robust principal component analysis based on maximum correntropy criterion. IEEE Trans. Image Process. 20(6), 1485–1494 (2011)MathSciNetCrossRefMATH
13.
Ke, Q., Kanade, T.: Robust \(L_{1}\) 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, 2005, vol. 1, pp. 739–746 (2005)
14.
Kwak, N.: Principal component analysis based on L1-norm maximization. IEEE Trans. Pattern Anal. Machine Intell. 30(9), 1672–1680 (2008)CrossRef
15.
Kwak, N.: Principal component analysis by \(l_{p}\)-norm maximization. IEEE Trans. Cyber. 44(5), 594–609 (2014)CrossRef
16.
Liang, Z., Xia, S., Zhou, Y., Zhang, L., Li, Y.: Feature extraction based on \(L_{p}\)-norm generalized principal component analysis. Pattern Recognit. Lett. 34(9), 1037–1045 (2013)CrossRef
17.
Ng, A.Y.: Feature selection, \(L_{1}\) vs. \(L_{2}\) regularization, and rotational invariance. In: Proceedings of the 21st International Conference on Machine Learning, ICML ’04 (2004)
18.
Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, New York (1981)MATH
19.
Liu, W., Pokharel, P.P., Principe, J.C.: Correntropy: properties and applications in non-gaussian signal processing. IEEE Trans. Signal Process. 55(11), 5286–5298 (2007)MathSciNetCrossRef
20.
Bullen, P.: Handbook of Means and Their Inequalities, 2nd edn. Kluwer Academic Publisher, Dordrecht (2003)CrossRefMATH
21.
Oh, J., Kwak, N.: Generalized mean for robust principal component analysis. Pattern Recognit. 54, 116–127 (2016)CrossRef
22.
Candès, E.J., Li, X., Ma, Y., Wright, J.: Robust Principal Component Analysis? J. ACM 58(3), 11:1–11:37 (2011)
23.
24.
Golub, G.H., Loan, C.F.V.: Matrix Computations, 3rd edn. Johns Hopkins, Baltimore (1996)MATH
25.
Oh, J., Kwak, N., Lee, M., Choi, C.H.: Generalized mean for feature extraction in one-class classification problems. Pattern Recognit. 46(12), 3328–3340 (2013)CrossRef
26.
Bishop, C.M.: Pattern Recongntion and Machine Learning. Springer, Berlin (2006)
27.
Comaniciu, D., Meer, P.: Mean shift: a robust approach toward feature space analysis. IEEE Trans. Pattern Anal. Machine Intell. 24(5), 603–619 (2002)CrossRef
28.
Phillips, P., Moon, H., Rizvi, S., Rauss, P.: The FERET evaluation methodology for face-recognition algorithms. IEEE Trans. Pattern Anal. Machine Intell. 22(10), 1090–1104 (2000)CrossRef
29.
LeCun, Y., Huang, F.J., Bottou, L.: Learning Methods for Generic Object Recognition with Invariance to Pose and Lighting. In: Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 2, pp. II–97–104 (2004)
30.
Hauberg, S., Feragen, A., Black, M.: Grassmann Averages for Scalable Robust PCA. In: Proceedings of the 2014 IEEE Conference on Computer Vision and Pattern Recognition, pp. 3810–3817 (2014)
Metadata
Title
Robust PCAs and PCA Using Generalized Mean
Authors
Jiyong Oh
Nojun Kwak
Copyright Year
2018
Publisher
Springer Singapore
Book
Advances in Principal Component Analysis

Print ISBN: 978-981-10-6703-7

Electronic ISBN: 978-981-10-6704-4

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-981-10-6704-4_4