Asymptotic level density in topological feature maps

The Kohonen algorithm entails a topology conserving mapping of an input pattern space X/spl sub/R/sup n/ characterized by an a priori probability distribution P(x), x/spl isin/X, onto a discrete lattice of neurons r with virtual positions w/sub rspl isin/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(W/sub r/) of the virtual net is a polynomial function of the probability density P(x) with D(w/sub r/)/spl sim/P/sup /spl alpha(w/sub r/). Here the distortion exponent is given by /spl alpha/=(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.<
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