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- E. Erwin, K. Obermayer, and
K. Schulten. Self-Organizing Maps: stationary States, Metastability and
Convergence Rate.
.
Biol. Cybern., 67:35-45, 1992.
(FTP Gzipped PostScript, 32 pages, 123 kb)
We investigate the effect of various types of neighborhood function
on the convergence rates and the presence or absence of metastable stationary
states of Kohonens self-organizing feature map algorithm in one dimension.
We demonstrate that the time necessary to form a topographic representation
of the unit interval [0, 1] may vary over several orders of magnitude
depending on the range and also the shape of the neighborhood function, by
which the weight changes of the neurons in the neighborhood of the winning
neuron are scaled. We will prove that for neighborhood functions which are
convex on an interval given by the length of the Kohonen chain there exist no
metastable states. For all other neighborhood functions, metastable states
are present and may trap the algorithm during the learning process. For the
widely-used Gaussian function there exists a threshold for the width above
which metastable states cannot exist. Due to the presence or absence of
metastable states, convergence time is very sensitive to slight changes in
the shape of the neighborhood function. Fastest convergence is achieved using
neighborhood functions which are "convex" over a large range
around the winner neuron and yet have large differences in value at
neighboring neurons.
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