Abstract
A formal analysis of the neighborhood interaction function selection in the topology preserving unsupervised neural network is presented in this paper. The definition of the neighborhood interaction function is motivated by anatomical evidence as opposed to what is currently used, which is a uniform neighborhood interaction set. By selecting a neighborhood interaction function with a neighborhood amplitude of interaction which is decreasing in spatial domain the topological order is always enforced and the rate of self-organization to final equilibrium state is improved. Several simulations are carried out to show the improvement in rate between using a neighborhood interaction function vs. using a neighborhood interaction set. An error measure functional is further defined to compare the two approaches quantitatively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carpenter GA, Grossberg S (1987) ART2: self-organization of stable catagory recognition codes for analog input patterns. Appl Opt 26:4919–4930.
Cottrell M, Fort JC (1986) A stochastic model of retinotopy: a self organizing process. Biol Cybern 53:405–411.
Fujita M, Bavarian B (1991) An ART2-TPM neural network for pattern classification. IEEE Trans Neural Net (submitted for publication).
Kohonen T (1982a) Self-organized formation of topologically correct feature maps. Biol Cybern 43:59–69.
Kohonen T (1982b) Clustering, taxonomy, and topological maps of patterns. Sixth Interational Conference on Pattern Recognition, Munich, Germany, October 19–22, 1982, pp 114–128.
Kohonen T (1982c) Analysis of simple self-organizing process. Biol Cybern 44:135–140.
Kohonen T (1984) Phonotopic maps — insightful representation of phonological feature for speech recognition. Proceedings of the Seventh International Conference on Pattern Recognition, Montreal, Canada, July 30–August 2, 1984, pp 114–128.
Kohonen T (1988) Self-organization and associative memory. Springer Series in Information Sciences. Springer, Berlin Heidelberg New York.
Lippmann P (1987) An introduction to computing with neural nets. IEEE ASSP Magazine 4–22.
Ritter H, Schulten K (1986a) Topology conserving mapping for learning motor tasks. AIP Conference proceeding 151, Snowbird Utah, pp 376–380.
Ritter H, Schulten K (1986b) Extending Kohonen's self-organizing mapping algorithm to learn ballistic movements. In: Eckmiller R, von der Malsburg C (eds) Neural computers. Springer, Berlin Heidelburg New York, pp 393–406.
Ritter H, Schulten K (1986c) On the stationary state of Kohonen's self-organizing sensory mapping. Biol Cybern 54:99–106.
Ritter H, Schulten K (1988) Convergence properties of Kohonen's topology conserving maps: fluctuations, stability, and dimension selection. Biol Cybern 60:59–71.
Takeuchi A, Amari S (1979) Formation of topographic maps and columnar microstructures in nerve fields. Biol Cybern 35:63–72.
Willshaw DJ, Malsburg C von der (1976) How patterned neural connections can be set up by self-organization. Proc R Soc London B194:431–445.
Willshaw DJ, Malsburg C von der (1979) A marker induction mechanism for the establishment of ordered neural mapping: its application to the retinotectal problem. Proc R Soc London B287:203–243.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lo, Z.P., Bavarian, B. On the rate of convergence in topology preserving neural networks. Biol. Cybern. 65, 55–63 (1991). https://doi.org/10.1007/BF00197290
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00197290