Asymptotic level density for a class of vector quantization processes

It is shown that for a class of vector quantization processes, related to neural modeling, that the asymptotic density Q(x ) of the quantization levels in one dimension in terms of the input signal distribution P(x) is a power law Q(x)=C-P(x)(alpha ), where the exponent alpha depends on the number n of neighbors on each side of a unit and is given by alpha=2/3-1/(3n (2)+3n+1(2)). The asymptotic level density is calculated, and Monte Carlo simulations are presented.