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- E. Erwin, K. Obermayer, and
K. Schulten. Self-Organizing Maps: ordering, Convergence Properties and
Energy Functions.
.
Biol. Cybern., 67:47-55, 1992.
(FTP Gzipped PostScript, 26 pages, 84 kb)
We investigate the convergence properties of the self-organizing
feature map algorithm for a simple, but very instructive case: the formation
of a topographic representation of the unit interval [0, 1] by a linear chain
of neurons. We extend the proofs of convergence of Kohonen and of Cottrell
and Fort to hold in any case where the neighborhood function, which is used
to scale the change in the weight values at each neuron, is a monotonically
decreasing function of distance from the winner neuron. We prove that the
learning dynamics cannot be described by a gradient descent on a single
energy function, but may be described using a set of potential functions, one
for each neuron, which are independently minimized following a stochastic
gradient descent. We derive the correct potential functions for the one- and
multi-dimensional case, and show that the energy functions given by Tolat
(1990) are an approximation which is no longer valid in the case of highly
disordered maps or steep neighborhood functions.
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