Code vector density in topographic mappings: Scalar case

The author derives some new results that build on his earlier work (1989) of combining vector quantization (VQ) theory and topographic mapping (TM) theory. A VQ model (with a noisy transmission medium) is used to model the processes that occur in TMs, which leads to the standard TM training algorithm, albeit with a slight modification to the encoding process. To emphasize this difference, the model is called a topographic vector quantizer (TVQ). In the continuum limit of the one-dimensional (scalar) TVQ. It is found that the density of code vectors is proportional to P(x)(a) (alpha=1/3) assuming that the transmission medium introduces additive noise with a zero-mean, symmetric, monotically decreasing probability density. This result is dramatically different from the result that is predicted when the standard TM training algorithm is used with a uniform symmetric neighborhood [-n, +n], and it is noted that this difference arises entirely from using minimum distortion rather than nearest neighbor encoding.