Abstract
We analyse a Markovian algorithm for the formation of topologically correct feature maps proposed earlier by Kohonen. The maps from a space of input signals onto an array of formal neurons are generated by a learning scheme driven by a random sequence of input samples. The learning is described by an equivalent Fokker-Planck equation. Convergence to an equilibrium map can be ensured by a criterion for the time dependence of the learning step size. We investigate the stability of the equilibrium map and calculate the fluctuations around it. We also study an instability responsible for a phenomenon termed by Kohonen “automatic selection of feature dimensions”.
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References
Bertsch H, Dengler J (1987) Klassifizierung und Segmentierung medizinischer Bilder mit Hilfe der selbstlernenden topologischen Karte. In: Paulus E (ed) 9. DAGM-Symposium Mustererkennung. Springer Informatik Fachberichte 149. Springer, Berlin Heidelberg New York, pp 166–170.
Cottrell M, Fort JC (1986) A stochastic model of retinotopy: a self-organizing process. Biol Cybern 53:405–411.
Erdi P, Barna G (1984) Self-organizing mechanism for the formation of ordered neural mappings. Biol Cybern 51:93–101.
Gardiner CW (1985) Handbook of stochastic methods, 2nd edn. Springer, Berlin Heidelberg New York.
Grossberg S (1976a) On the development of feature detectors in the visual cortex with applications to learning and reactiondiffusion systems. Biol Cybern 21:145–159.
Grossberg S (1976b) Adaptive pattern classification and universal recoding: I. Parallel development and coding of neural feature detectors. Biol Cybern 23:121–134.
Kaas JH, Merzenich MM, Killackey HP (1983) The reorganization of somatosensory cortex following peripheral nerve damage in adult and developing mammals. Ann Rev Neurosci 6:325–356.
Kampen NG van (1981) Stochastic processes in physics and chemistry. North Holland, Amsterdam.
Knudsen EI, du Lac S, Esterly SD (1987) Computational maps in the brain. Ann Rev Neurosci 10:41–65.
Kohonen T (1982a) Self-organized formation of topologically correct feature maps. Biol Cybern 43:59–69.
Kohonen T (1982b) Analysis of a simple self-organizing process. Biol Cybern 44:135–140.
Kohonen T (1982c) Clustering, taxonomy and topological maps of patterns. Proceedings of the 6th International Conference on Pattern Recognition, Munich. pp 114–128.
Kohonen T (1984) Self-organization and associative memory. Springer Series in Information Sciences 8. Springer, Berlin Heidelberg New York.
Kohonen T (1986) Learning vector quantization for pattern recognition. Helsinki University of Technology, Report TKK-F-A601.
Kohonen T, Mäkisara K, Saramäki T (1984) Phonotopic maps, — insightful representation of phonological features for speech recognition. Proceedings of the 7th International Conference on Pattern Recognition, Montreal. pp 182–185.
Kushner HJ, Clark DS (1978) Stochastic approximation methods for constrained and unconstrained systems. Springer, Berlin Heidelberg New York.
Malsburg C von der (1979) Development of ocularity domains and growth behaviour of axon terminals. Biol Cybern 32:49–62.
Overton KJ, Arbib MA (1982) The branch arrow model of the formation of retino-tectal connections. Biol Cybern 45:157–175.
Ritter H, Schulten K (1986a) On the stationary state of Kohonen's self-organizing sensory mapping. Biol Cybern 54:99–106.
Ritter H, Schulten K (1986b) Topology conserving mappings for learning motor tasks. In: Denker JS (eds) Neural networks of computing. AIP Conference Proceedings 151. Snowbird, Utah, pp 376–380.
Ritter H, Schulten K (1987) Extending Kohonen's self-organizing mapping algorithm to learn ballistic movements. In: Eckmiller R, von der Malsburg C (eds) Neural computers. Springer, Berlin Heidelberg New York, pp 393–406.
Suga N, O'Neill WE (1979) Neural axis representing target range in the auditory cortex of the mustache Bat Sci 206:351–353.
Takeuchi A, Amari S (1979) Formation of topographic maps and columnar microstructures. 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 B 194:431–445.
Willshaw DJ, Malsburg C von der (1979) A marker induction mechanism for the establishment of ordered neural mappings: its application to the retinotectal problem. Proc R Soc London B 287:203–243.
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Ritter, H., Schulten, K. Convergence properties of Kohonen's topology conserving maps: fluctuations, stability, and dimension selection. Biol. Cybern. 60, 59–71 (1988). https://doi.org/10.1007/BF00205972
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DOI: https://doi.org/10.1007/BF00205972