Edgar Erwin, Klaus Obermayer, and Klaus Schulten.
Self-organizing maps: Ordering, convergence properties and energy
functions.
Biological Cybernetics, 67:47-55, 1992.
ERWI92A
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one- and multi-dimensional case, and show that the energy functions given by Tolat (1990) are an approximation which is no longer valid in the case of highly disordered maps or steep neighborhood functions.
Download Full Text
The manuscripts available on our site are provided for your personal
use only and may not be retransmitted or redistributed without written
permissions from the paper's publisher and author. You may not upload any
of this site's material to any public server, on-line service, network, or
bulletin board without prior written permission from the publisher and
author. You may not make copies for any commercial purpose. Reproduction
or storage of materials retrieved from this web site is subject to the
U.S. Copyright Act of 1976, Title 17 U.S.C.