The effect of concave and convex weight adjustments on self-organizing maps

Two nonlinear models of weight adjustments of self-organizing maps are derived to obtain desirable densities of output units, one that approaches the probability distribution p(ξ) of the inputs and one that approaches a uniform distribution. If a convex model is used to adjust weights, the density of output units can be made to approach p(ξ) instead of the p(ξ)^2/3 which results from the linear weight adjustment of Kohonen's self-organizing maps. If a concave model of weight adjustments is used, the density approaches a uniform distribution and the winner frequency distribution of output units is proportional to p(ξ). The former can provide more efficient data representations for vector quantization, while the latter can provide more meaningful measures for cluster analysis. Numerical demonstrations validate the mathematical derivations. The convergence of the concave model is faster than the linear and convex models while the convergence of the convex model is comparable to that of the linear model.