Decision region approximation by polynomials or neural networks

We give degree of approximation results for decision regions which are defined by polynomial and neural network parametrizations. The volume of the misclassified region is used to measure the approximation error, and results for the degree of L₁ approximation of functions are used. For polynomial parametrizations, we show that the degree of approximation is at least 1, whereas for neural network parametrizations we prove the slightly weaker result that the degree of approximation is at least r, where r can be any number in the open interval (0, 1)