APPROXIMATION OF FUNCTIONS ON A COMPACT SET BY FINITE SUMS OF A SIGMOID FUNCTION WITHOUT SCALING

This paper is concerned with three layered feedforward neural networks which are capable of approximately representing continuous functions on a compact set. Of particular interest here is the use of a sigmoid function without scaling. First, we prove existentially that a linear combination of unscaled shifted rotations of any sigmoid function can approximate uniformly an arbitrary continuous function on a compact set in R(d). Second, a proposition is proved constructively using the fact that a homogeneous polynomial P can be expressed as P(x) = SIGMA-i = 1n a(i)(omega-i.x)r. It states that the approximation of an arbitrary polynomial on a interval in R can be extended to that of an arbitrary continuous function on a compact set in R(d). Then, four corollaries are derived. Though their statements are more or less restricted, the proofs provide algorithms for implementing the uniform approximation. In three of these corollaries, sigmoid functions are used without scaling.