Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T06:05:38.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence in distribution of the multi-dimensional Kohonen algorithm

Part of: Mathematical biology in general Markov processes

Published online by Cambridge University Press:  14 July 2016

Ali A. Sadeghi
Ali A. Sadeghi*
Affiliation:
University of Calgary
*
∗ Postal address: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.

Keywords

neural networksmulti-dimensional Kohonen algorithmMarkov processstochastic stabilityuniform ergodicity

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 92B20: Neural networks, artificial life and related topics
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 136 - 151
Copyright
Copyright © by the Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benaim, M., Fort, J. C. and Pagés, G. (1998). Convergence of the one-dimensional Kohonen algorithm. Adv. Appl. Prob. 30, 850–869.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Bouton, C. and Pagés, G. (1997). About the multidimensional competitive learning vector quantization algorithm with constant gain. Ann. Appl. Prob. 7, 679–710.Google Scholar
Burton, R. M., and Faris, W. G. (1996). A self-organizing cluster process. Ann. Appl. Prob. 6, 1232–1247.Google Scholar
Cottrell, M., and Fort, J. C. (1987). Étude d'un processus d'auto-organisation. Ann. Inst. H. Poincaré 23, 1–20.Google Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Erwin, E., Obermeyer, K., and Schulten, K. (1992). Self-organizing maps: ordering, convergence properties and energy functions. Biol. Cybernet. 67, 47–55.CrossRefGoogle ScholarPubMed
Fang, H., Gong, G., and Qian, M. (1997). Annealing of iterative stochastic schemes. SIAM J. Control Optim. 35, 1886–1907.CrossRefGoogle Scholar
Flanagan, J. A. (1996). Self-organisation in Kohonen's SOM. Neural Networks 7, 1185–1197.Google Scholar
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.Google Scholar
Fort, J. C. and Pagés, G. (1996). About the Kohonen algorithm: strong or weak self-organization? Neural Networks 5, 773–785.Google Scholar
Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biol. Cybernet. 43, 59–69.CrossRefGoogle Scholar
Kohonen, T. (1995). Self-organizing Maps. Springer, Berlin.CrossRefGoogle Scholar
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Sadeghi, A. A. (1998). Asymptotic behavior of self-organizing maps with non-uniform stimuli distribution. Ann. Appl. Prob. 8, 281–299.Google Scholar
Sadeghi, A. A. (1998). Self-organization property of Kohonen's map with general type of stimuli distribution. Neural Networks 11, 1637–1643.Google Scholar
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.Google Scholar
Sadeghi, A. A. (1999). A Mathematical Study of Self-organizing Neural Networks. Shaker, Aachen.Google Scholar
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.CrossRefGoogle Scholar