A universal mapping for kolmogorov's superposition theorem

✍ Scribed by David A. Sprecher

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 414 KB
• Volume: 6
• Category: Article
• ISSN: 0893-6080
• DOI: 10.1016/s0893-6080(09)80020-8

✦ Synopsis

Based on constructions of Kolmogorov and an earlier refinement 0/ the author, we lise a sequence of integrally independent positive numbers to construct a continuousfunction -./I( x) having thefollowing property: Every real-valued uniformly continuous function f( XI • . . . • x n) of n ~2 variables can be obtained as a superposition of continuousfunctions ofone variable based on weighted sums oftranslates cfthefixedfun aion !J;( x) that is independent ofthe number ofvariables n. From th is is obtained a stronger version ofth« Hecht-Nielsen three-layer[eedforward neural network/or implementing f( XI • . . . , XII)' Keywords-Superpositions, Ko1mogorov, Representations of continuous functions of several variables, Feedforward neural networks, Processing elements, Weighted sums, Uniform continuity. (I) (2)

In what follows, :R designates the real line, t;

and:ll n and t;n designate the respective n-fold Cartesian products of:R and t;. Continuity and convergence are uniform in the Euclidean metric.

The Superposition Theorem ofKolmogorov ( 1957) establishes that for each integer n ~2 there are n X (2n + 1) continuous monotonically increasing functions YJpq with the following property: For every real-