Neural network approximation of continuous functionals and continuous functions on compactifications

✍ Scribed by M.B. Stinchcombe

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 164 KB
• Volume: 12
• Category: Article
• ISSN: 0893-6080
• DOI: 10.1016/s0893-6080(98)00108-7

✦ Synopsis

This article characterizes the set of activation functions, bounded or unbounded, that allow feedforward network approximation of the continuous functions on the classic two-point compactification of R 1 . The characterization fails when the set of targets are continuous functions on the classic compactifications of R n , n Ն 2. Nonpolynomial, analytic activation functions with input-to-hidden weights in very limited sets allow approximation of continuous function over compact sets in R n , while even sigmoidal activation functions with weights in limited sets cannot approximate continuous functions on compactifications. The abstract structure foregrounded by compactification leads directly to possibility results for multi-layer networks and possibility results for neural networks in infinite dimensional settings.