Approximation of continuous functions on Rd by linear combinations of shifted rotations of a sigmoid function with and without scaling

✍ Scribed by Yoshifusa Ito

• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 909 KB
• Volume: 5
• Category: Article
• ISSN: 0893-6080
• DOI: 10.1016/s0893-6080(05)80009-7

✦ Synopsis

Abslracl--Let f be an arbitrary continuous function defined on R d and h be any sigmoid function. Then, there is a linear combination of scaled shifted rotations of h, which approximates f uniformly on the whole space R d, if a certain such linear combination approximates f uniformly in a neighbourhood of the infinite point. From this result, derived is that any function continuous on R d (the one-point compactification of R a) can be likewise approximated. Further, obtained is a necessary and sufficient condition on h, under which the uniform approximation can be implemented without scaling of h. The Heaviside function, the logistic function, the Gaussian distribution function, and the arctangent sigmoid function satisfy this condition. Any sigmoid function that increases only in a finite interval satisfies it. Examples of sigmoid functions incapable of implementing the uniform approximation without scaling are also illustrated.