Ridge Functions, Sigmoidal Functions and Neural Networks

This paper considers mainly approximation by ridge functions. Fix a point a 2 IR n and a function g : IR ! IR. Then the function f : IR n ! IR deened by f (x) = g(ax), x 2 IR n , is a ridge or plane wave function. A sigmoidal function is a particular example of the function g which closely resembles 1 at 1 and 0 at ?1. This paper discusses approximation problems involving general ridge functions and speciic research connected with sigmoidal functions. The type of problems discussed lead naturally to a consideration of neural networks, particularly multi-layered feedforward networks. Most important is the existence of constructive proofs of the fact that networks of this type can approximate a given continuous function to any desired accuracy. A mathematician's view of these networks may be found in Section 5.