Nonparametric density estimation and regression achieved with a learning rule for equiprobabilistic topographic map formation

An "online" learning rule, called the vectorial boundary adaptation rule (VBAR), is proposed for topographic map formation. Since VBAR is aimed at achieving all equiprobabilistic quantization of the input space, the weight density at convergence will be proportional to the input density. In this way, the converged map yields a nonparametric model of the input density. We use an information-theoretic measure (mutual information) to assess and compare the performance of VBAR with Kohonen's SOM algorithm. Finally, we show that topographic map formation with VBAR in "batch" mode is equivalent to statistical kernel smoothing (nonparametric regression).