Asymptotic level density in topological feature maps

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 w_r∈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_r) of the virtual net is a polynomial function of the probability density P(x) with D(w_r)~P^α(w_r). 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