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)(alpha), where P(x) is the input data density and alpha 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].