Convergence in distribution of the multi-dimensional Kohonen algorithm

Journal of Applied Probability

Volumn 38, Issue 1, 2001, Pages 136-151

  Sadeghi, Ali A ^a  

^a UNIVERSITY OF CALGARY   (Canada)

Author keywords

Markov process; Multi dimensional Kohonen algorithm; Neural networks; Stochastic stability; Uniform ergodicity

Indexed keywords

EID: 0035533776     PISSN: 00219002     EISSN: None     Source Type: Journal    
DOI: 10.1239/jap/996986649     Document Type: Article

References (20)

1. BENAIM, M., FORT, J. C. AND PAGÉS, G. (1998). Convergence of the one-dimensional Kohonen algorithm. Adv. Appl. Prob. 30, 850-869.
2. BOUTON, C. AND PAGÉS, G. (1993). Self-organization and a.s. convergence of the one-dimensional Kohonen algorithm with non-uniformly distributed stimuli. Stoch. Proc. Appl. 47, 249-274.
3. BOUTON, C. AND PAGÉS, G. (1994). Convergence in distribution of the one-dimensional Kohonen algorithm when the stimuli are not uniform. Adv. Appl. Prob. 26, 80-103.
4. BOUTON, C. AND PAGÉS, G. (1997). About the multidimensional competitive learning vector quantization algorithm with constant gain. Ann. Appl. Prob. 7, 679-710.
5. BURTON, R. M. AND FARIS, W. G. (1996). A self-organizing cluster process. Ann. Appl. Prob. 6, 1232-1247.
6. COTTRELL, M. AND FORT, J. C. (1987). Étude d'un processus d'auto-organisation. Ann. Inst. H. Poincaré 23, 1-20.
7. DOOB, J. L. (1953). Stochastic Processes. John Wiley, New York.
8. ERWIN, E., OBERMEYER, K. AND SCHULTEN, K. (1992). Self-organizing maps: ordering, convergence properties and energy functions. Biol. Cybernet. 67, 47-55.
9. FANG, H., GONG, G. AND QIAN, M. (1997). Annealing of iterative stochastic schemes. SIAM J. Control Optim. 35, 1886-1907.
10. FLANAGAN, J. A. (1996). Self-organisation in Kohonen's SOM. Neural Networks 7, 1185-1197.
11. FORT, J. C. AND PAGÉS, G. (1995). On the a.s. convergence of the Kohonen algorithm with a general neighborhood function. Ann. Appl. Prob. 4, 1177-1216.
12. FORT, J. C. AND PAGÉS, G. (1996). About the Kohonen algorithm: strong or weak self-organization? Neural Networks 5, 773-785.
13. KOHONEN, T. (1982). Self-organized formation of topologically correct feature maps. Biol. Cybernet. 43, 59-69.
14. KOHONEN, T. (1995). Self-organizing Maps. Springer, Berlin.
15. MEYN, S. P. AND TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
16. SADEGHI, A. A. (1998). Asymptotic behavior of self-organizing maps with non-uniform stimuli distribution. Ann. Appl. Prob. 8, 281-299.
17. SADEGHI, A. A. (1998). Self-organization property of Kohonen's map with general type of stimuli distribution. Neural Networks 11, 1637-1643.
18. SADEGHI, A. A. (1998). Self-organization and convergence of the one-dimensional Kohonen algorithm. In Proc. 6th Eur. Symp. Artificial Neural Networks, Bruges. D-Facto, Brussels, pp. 173-178.
19. SADEGHI, A. A. (1999). A Mathematical Study of Self-organizing Neural Networks. Shaker, Aachen.
20. VILLMANN, T., DER, R., HERRMANN, M. AND MARTINETZ, T. (1997). Topology preservation in self-organizing feature maps: exact definition and measurement. IEEE Trans. Neural Networks 2, 256-266.