A Numerical Implementation of Kolmogorov's Superpositions II
This paper presents a numerical algorithm for the parallel computations ?q in the Kolmogorov superpositions thereby providing the final step in their numerical implementation. The first step consisting of the f-independent computation of the functions ?(xq) = ?pn=1 ?p?(xp + qa) with a fixed ? and constants a and ?p in a hidden layer in the Hecht-Nielsen feedforward neural network has been accomplished previously. The step taken in this paper is the implementation of the output layer of the network that computes an arbitrary known continuous real-valued function f defined on the unit cube n. Employed for the purpose is an iterative method which is intended as a basis for the possible development of adaptive methods that build on this approach. Each function ?q is obtained iteratively through a series ?r?qr which is determined on an f and q dependent subsequence  of rational coordinates such that ?qr is determined at the coordinate points The paper also includes alternative constructions of the functions ?qr and a brief discussion of the differentiability of ?q ? ?(xq); together with a previous result it gives a constructive proof of Kolmogorov's theorem.  1997 Elsevier Science Ltd. All Rights Reserved.
kolmogorov superpositions functions continuity computation of multivariate functions approximation of multivariate representation of multivariate functions