Geometrical Aspects of Discrimination by Multilayer Perceptrons

We investigate some geometrical aspects of the discriminant functions of the kindfp(x)=∑pk=1 ckτ(a′kx) for suitable constantsak, ckwhereτis a sigmoidal transformation. This function is realized by a multilayer perceptron with one hidden layer. These results are applied in the analysis of the discriminating power offp. In particular, we prove that the class of finite populationsΩ1andΩ2that can be distinguished byfpis monotonically increasing inpand we give a minimal sufficientpleading to a complete separation betweenΩ1andΩ2.