Asymptotic level density in topological feature maps

Authors:
D. R. Dersch;P. Tavan
Affiliations:
Inst. fur Medizinische Optik, Theor. Biophys., Ludwig-Maximilians-Univ., Munchen;-
Venue:
IEEE Transactions on Neural Networks
Year:
1995

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Abstract

The Kohonen algorithm entails a topology conserving mapping of an input pattern space X⊂Rn characterized by an a priori probability distribution P(x), x∈X, onto a discrete lattice of neurons r with virtual positions wr∈X. Extending results obtained by Ritter (1991) the authors show in the one-dimensional case for an arbitrary monotonously decreasing neighborhood function h(|r-r'|) that the point density D(Wr) of the virtual net is a polynomial function of the probability density P(x) with D(wr)~Pα(wr). Here the distortion exponent is given by α=(1+12R)/3(1+6R) and is determined by the normalized second moment R of the neighborhood function. A Gaussian neighborhood interaction is discussed and the analytical results are checked by means of computer simulations