A Better Approximation for Balls Original Research Article

Unexpectedly accurate and parsimonious approximations for balls in Rd and related functions are given using half-spaces. Instead of a polytope (an intersection of half-spaces) which would require exponentially many half-spaces (of order (1ε)d) to have a relative accuracy ε, we use T=c(d2/ε2) pairs of indicators of half-spaces and threshold a linear combination of them. In neural network terminology, we are using a single hidden layer perceptron approximation to the indicator of a ball. A special role in the analysis is played by probabilistic methods and approximation of Gaussian functions. The result is then applied to functions that have variation Vf with respect to a class of ellipsoids. Two hidden layer feedforward sigmoidal neural nets are used to approximate such functions. The approximation error is shown to be bounded by a constant times Vf/T1/21+Vf d/T1/42, where T1 is the number of nodes in the outer layer and T2 is the number of nodes in the inner layer of the approximation fT1, T2.