Convergence analysis for principal component flows

Shintaro Yoshizawa; Uwe Helmke; Konstantin Starkov

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 1, page 223-236
  • ISSN: 1641-876X

Abstract

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A common framework for analyzing the global convergence of several flows for principal component analysis is developed. It is shown that flows proposed by Brockett, Oja, Xu and others are all gradient flows and the global convergence of these flows to single equilibrium points is established. The signature of the Hessian at each critical point is determined.

How to cite

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Yoshizawa, Shintaro, Helmke, Uwe, and Starkov, Konstantin. "Convergence analysis for principal component flows." International Journal of Applied Mathematics and Computer Science 11.1 (2001): 223-236. <http://eudml.org/doc/207501>.

@article{Yoshizawa2001,
abstract = {A common framework for analyzing the global convergence of several flows for principal component analysis is developed. It is shown that flows proposed by Brockett, Oja, Xu and others are all gradient flows and the global convergence of these flows to single equilibrium points is established. The signature of the Hessian at each critical point is determined.},
author = {Yoshizawa, Shintaro, Helmke, Uwe, Starkov, Konstantin},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Hessians; neural networks; principal component analysis; phase portrait; gradient flows},
language = {eng},
number = {1},
pages = {223-236},
title = {Convergence analysis for principal component flows},
url = {http://eudml.org/doc/207501},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Yoshizawa, Shintaro
AU - Helmke, Uwe
AU - Starkov, Konstantin
TI - Convergence analysis for principal component flows
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 1
SP - 223
EP - 236
AB - A common framework for analyzing the global convergence of several flows for principal component analysis is developed. It is shown that flows proposed by Brockett, Oja, Xu and others are all gradient flows and the global convergence of these flows to single equilibrium points is established. The signature of the Hessian at each critical point is determined.
LA - eng
KW - Hessians; neural networks; principal component analysis; phase portrait; gradient flows
UR - http://eudml.org/doc/207501
ER -

References

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  2. Baldi P. and Hornik K. (1995): Learning in linear neural networks: A survey. - IEEE Trans. Neural Netw., Vol.6, No.4, pp.837-858. 
  3. Brockett R.W. (1991): Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems. - Lin. Algebra Appl., Vol.146, pp.79-91. Zbl0719.90045
  4. Helmke U. and Moore J.B. (1994): Dynamical Systems and Optimization. - London: Springer. Zbl0984.49001
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  10. Oja E., Ogawa H. and Wangviwattana J. (1992b): Principal component analysis by homogeneous neural networks, Part II: Analysis and extensions of the learning algorithms. - IEICE Trans. Inf. Syst., Vol.3, pp.376-382. 
  11. Sanger T.D. (1989): Optimal unsupervised learning in a single-layer linear feedforward network. - Neural Netw., Vol.2, No.6, pp.459-473. 
  12. Williams R. (1985): Feature discovery through error-correctinglearning. - Tech. Rep. No.8501, University of California, San Diego, Inst. of Cognitive Science. 
  13. Wyatt J.L. and Elfadel I.M. (1995): Time-domain solutions of Oja's equations. - Neural Comp.,Vol.7, No.5, pp.915-922. 
  14. Xu L. (1993): Least mean square error recognition principle for self organizing neural nets. - Neural Netw., Vol.6, No.5, pp.627-648. 
  15. Yan W.Y., Helmke U. and Moore J.B. (1994): Global analysis of Oja's flow for neural networks. - IEEE Trans. Neural Netw., Vol.5, No.5, pp.674-683. 

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