Code Vector Density in Topographic Mappings
Scalar Case · 1991
In coding theory one transforms signals from a source representation into an encoded representation that is suitable for transmission through a (possibly noisy) medium, and upon reception one decodes to reconstruct an approxima- tion to the original signal. In autoassociative network theory one reconstructs a signal given incomplete (and possibly noisy) information about it. We derive some new results by combining these two approaches in the form of vector quantization (VQ) theory and topographic mapping (TM) theory. We use a VQ model (with a noisy transmission medium) to model the pro- cesses that occur in TM's, which leads to the standard TM training algorithm, albeit with a slight modification to the en- coding process (minimum distortion rather than nearest neigh- bor encoding). To emphasize this difference we call our model a topographic vector quantizer (TVQ). In the continuum limit of the one-dimensional (scalar) TVQ we find that the density of code vectors is proportional to (a = 1/3) (which is the same as the result obtained from a standard scalar quantizer), assuming that the transmission medium introduces additive noise with a zero-mean, symmetric, monotonically decreasing probability density (which is equivalent to using a symmetri- cally tapered neighborhood in a TM). Our a = 1/3 result is dramatically different from the a = ((2n + 1)'/3)/((n + l)* + n2) result that is predicted when the standard TM training al- gorithm is used with a uniform symmetric neighborhood ( -n, +a), and we note that this difference arises entirely from using minimum distortion rather than nearest neighbor encoding. We verify our new result by performing a numerical experiment using P(x) a x.