TY - JOUR

T1 - Self-organizing maps

T2 - ordering, convergence properties and energy functions

AU - Erwin, E.

AU - Obermayer, K.

AU - Schulten, K.

PY - 1992/5

Y1 - 1992/5

N2 - 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 oneand multi-dimensional case, and show that the energy functions given by Tolat (1990) are an approximation which is no longer valid in the case of highly disordered maps or steep neighborhood functions.

AB - 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 oneand multi-dimensional case, and show that the energy functions given by Tolat (1990) are an approximation which is no longer valid in the case of highly disordered maps or steep neighborhood functions.

UR - http://www.scopus.com/inward/record.url?scp=0026436795&partnerID=8YFLogxK

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U2 - 10.1007/BF00201801

DO - 10.1007/BF00201801

M3 - Article

C2 - 1606243

AN - SCOPUS:0026436795

VL - 67

SP - 47

EP - 55

JO - Biological Cybernetics

JF - Biological Cybernetics

SN - 0340-1200

IS - 1

ER -