A weak condition on linear independence of unscaled shifts of a function and finite mappings by neural networks

Abstract

Let 1 ≤ c ≤ d and let g be a slowly increasing function defined on Rc. Suppose that the support of the Fourier transform Fcg of g includes a converging sequence of distinct points yκ which sufficiently rapidly come close to a line as κ→ ∞. Then, any mapping of any number, say n, of any points x1, xn in Rd onto R can be implemented by a linear sum of the form ∑ n j=1 ajg(Wxi + zj ). Here, W is a d × c matrix having orthonormal row vectors, implying that g is used without scaling, and that the sigmoid function defined on R and the radial basis function defined on Rd are treated on a common basis. © Springer-VerlagBerlin Heidelberg 2002.