ASYMPTOTIC LEVEL DENSITY IN TOPOLOGICAL FEATURE MAPS

The Kohonen algorithm entails a topology conserving mapping of an input pattern space X subset of R(n) characterized by an a priori probability distribution P(x), x is an element of X, onto a discrete lattice of neurons tau with virtual positions w(tau)is an element of X. Extending results obtained by Ritter [4] we show in the one-dimensional case for an arbitrary monotonously decreasing neighborhood function h(\tau - tau'\) that the point density D(w(tau)) of the virtual net is a polynomial function of the probability density P(x) with D(w(tau)) similar to P-alpha(w(tau)). Here the distortion exponent is given by 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.