Codeword distribution for frequency sensitive competitive learning with one-dimensional input data

We study the codeword distribution for a conscience-type competitive learning algorithm, frequency sensitive competitive learning (FSCL), using one-dimensional input data. We prove that the asymptotic codeword density in the limit of large number of codewords is given by a power law of the form Q(x)=C·P(x)^α, where P(x) is the input data density and α depends on the algorithm and the form of the distortion measure to be minimized. We further show that the algorithm can be adjusted to minimize any L_p distortion measure with p ranging in (0,2]