Abstract
In this paper we present a detailed investigation of the statistical and convergence properties of the Kohonen's Self-Organising Mapping (SOM) in any dimension. Using an extend Central Limit Theorem, we prove that the feature space in SOM learning is an approximation to multiple Gausssian distributed stochastic processes, which will eventually converge in the meansquare sense to the density centres of the input probabilistic sub-spaces. We also demonstrate that combining the SOM with a Kalman filter can smooth and accelerate the learning and convergence of the SOM. In our applications, we show that such a modified SOM achieves a much better performance, namely a lower distortion than the original algorithm, especially in early training stages, and at low extra computational cost. This modification will be particular useful when the available training set is small.
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References
Allinson, N. M. 1990. Self-organising maps and their applications. Theory and Applications of Neural networks. J. G. Taylor and C. L. T. Mannion eds. London, Springer-Verlay. 101–118.
Bauer, H.-U., and Pawelzik, K. R. 1992. Quantifying the neighborhood preservation of self-organizing feature maps. IEEE Trans. Neural Networks 3(4), 570–579.
Cho, C.-M., and Don, H.-S. 1991. A parallel Kalman algorithm for fast learning of multilayer neural networks. Proc. IJCNN-91 (Singapore), 2044–2049.
Erwin, E., Obermayer, K., and Schulten, K. 1992a. Self-organizing maps: ordering, convergence properties and energy functions. Biol. Cybern. 67, 47–55.
-1992b. Self-organizing maps: stationary states, metastability and convergence rate. Biol. Cybern. 67, 35–45.
Kohonen, T. 1982. Self-organized formation of topologically correct feature maps. Biol. Cybern. 43, 59–69.
-1984. Self-Organization and Associative Memory. Springer-Verlay.
-1991. Self-Organizing Maps: Optimization Approaches. Artificial Neural Networks. Elsevier. 981–990.
Lo, Z. P., and Bavarian, B. 1991. On the rate of convergence in topology preserving neural network. Biol. Cybern. 65, 55–63.
Ritter, H. and Schulten, K. 1988. Convergence properties of Kohonen's topology conserving maps: Fluctuations, stability, and dimension selection. Biol. Cybern. 60. 59–71.
Ruck, D. W., Rogers, S. K., Kabrisky, M., Maybeck, P. S., and Oxley, M. E. 1992. Comparative analysis of backpropagation and the extended Kalman filter for training multilayer perceptrons. IEEE Trans. PAMI-14(6), 686–691.
Yin, H. and Allinson, N. M. 1992. Stochastic Analysis and Treatment of the Kohonen's Self-Organising Map.
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© 1993 Springer-Verlag Berlin Heidelberg
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Yin, H., Allinson, N.M. (1993). On the distribution of feature space in Self-Organising mapping and convergence accelerating by a Kalman algorithm. In: Mira, J., Cabestany, J., Prieto, A. (eds) New Trends in Neural Computation. IWANN 1993. Lecture Notes in Computer Science, vol 686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56798-4_162
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DOI: https://doi.org/10.1007/3-540-56798-4_162
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