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Transition times in self-organizing maps

Abstract

We study the creation of topological maps. It is well known that topological defects, like kinks in one-dimensional maps or twists (‘butterflies’) in two-dimensional maps, can be (metastable) fixed points of the learning process. We are interested in transition times from these disordered configurations to the perfectly ordered configurations, i.e., the average time it takes to remove a kink or to unfold a twist. For this study we consider a self-organizing learning rule which is equivalent to the Kohonen learning rule, except for the determination of the ‘winning’ unit. The advantage of this particular learning rule is that it can be derived from an error potential. The existence of an error potential facilitates a global description of the learning process. Mappings in one and two dimensions are used as examples. For small lateral-interaction strength, topological defects correspond to local minima of the error potential, whereas global minima are perfectly ordered configurations. Theoretical results on the transition times from the local to the global minima of the error potential are compared with computer simulations of the learning rule.

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References

  1. Bauer H-U, Pawelzik KR (1992) Quantifying the neighborhood preservation of self-organizing feature maps. IEEE Trans Neural Networks 3:570–579

  2. Cottrell M, Fort JC (1987) A stochastic model of retinotopy: a self-organizing process. Biol Cybern 53:405–411

  3. Durbin R, Mitchison G (1990) A dimension reduction framework for understanding cortical maps. Nature 343:644–647

  4. Erwin E, Obermayer K, Schulten K (1992) Self-organizing maps: ordering, convergence properties and energy functions. Biol Cybern 67:47–55

  5. Geszti T (1990) Physical models of neural networks. World Scientific, Singapore

  6. Heskes TM (1994) On Fokker-Planck approximations of on-line learning processes. J Phys A 27:5145–5160

  7. Heskes TM, Kappen B (1991) Learning processes in neural networks. Phys Rev A 44:2718–2726

  8. Heskes TM, Kappen B (1993) Error potentials for self-organization. In: International Conference on Neural Networks, San Francisco, vol III. IEEE, New York, pp 1219–1223

  9. Heskes TM, Slijpen ETP, Kappen B (1992) Learning in neural networks with local minima. Phys Rev A 46:5221–5231

  10. Heskes TM, Slijpen ETP, Kappen B (1993) Cooling schedules for learning in neural networks. Phys Rev E 47:4457–4464

  11. Kohonen T (1982) Self-organized formation of topologically correct feature maps. Biol Cybern 43:59–69

  12. Kohonen T (1988) Self-organization and associative memory. Springer, Berlin Heidelberg New York

  13. Kohonen T (1991) Self-organizing maps: optimization approaches. In: Kohonen T etal, (eds) Artificial neural networks, vol II. North-Holland, Amsterdam, pp 981–990

  14. Luttrell SP (1989) Self-organisation: a derivation from first principles of a class of learning algorithms. In: International Joint Conference on Neural Networks, vol II. IEEE Computer Society Press, pp 495–498

  15. Luttrell SP (1994) A Bayesian analysis of self-organizing maps. Neural Comput 6:767–794

  16. Miller K, Keller J, Stryker M (1989) Ocular dominance column development: analysis and simulation. Science 245:605–615

  17. Obermayer K, Ritter H, Schulten K (1990) A principle for the formation of the spatial structure of cortical feature maps. Proc Natl Acad Sci USA 87:8345–8349

  18. Obermayer K, Blasdel G, Schulten K (1992) Statistical-mechanical analysis of self-organization and pattern formation during the development of visual maps. Phys Rev A 45:7568–7589

  19. Ritter H, Schulten K (1986) On the stationary state of Kohonen's self-organizing sensory mapping. Biol Cybern 54:99–106

  20. Ritter H, Schulten K (1988) Convergence properties of Kohonen's topology conserving maps: fluctuations, stability, and dimension selection. Biol Cybern 60:59–71

  21. Ritter H, Obermayer K, Schulten K, Rubner J (1991) Self-organizing maps and adaptive filters. In: Domany E etal (eds) Models of neural networks. Springer, Berlin Heidelberg New York, pp 281–306

  22. Rose K, Gurewitz E, Fox GC (1990) Statistical mechanics of phase transitions in clustering. Phys Rev Lett 65:945–948

  23. Takeuchi A, Amari S (1979) Formation of topographic maps and columnar microstructures. Biol Cybern 35:63–72

  24. Tolat VV (1990) An analysis of Kohonen's self-organizing maps using system of energy functions. Biol Cybern 64:155–164

  25. Van Kampen NG (1981) Stochastic processes in physics and chemistry. North-Holland, Amsterdam

  26. Von der Malsburg Ch (1973) Self-organization of orientation sensitive cells in the striate cortex. Kybernetik 14:85–100

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Heskes, T.M. Transition times in self-organizing maps. Biol. Cybern. 75, 49–57 (1996). https://doi.org/10.1007/BF00238739

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Keywords

  • Computer Simulation
  • Local Minimum
  • Learning Process
  • Theoretical Result
  • Global Minimum