On a constructive proof of Kolmogorov's superposition theorem
type: Item
instance of: scholarly article
title: On a constructive proof of Kolmogorov's superposition theorem
zbMATH Open document ID: 1194.26020
author: Jürgen Braun
author: Michael Griebel
published in: Constructive Approximation
publication date: 2010-01-15T00:00:00Z
review text: In 1957 Kolmogorov showed that any continuous function \(f:[0,1]^n\to \mathbb{R}\) can be represented as a composition of addition and some functions of one variable. His proof was not constructive. Recently Kolmogorov's superposition theorem was applied in the study of neural networks. Motivated by these applications, Sprecher [Neural Netw. 9, No. 5, 765--772 (1996); Neural. Netw. 10, No. 3, 447--457 (1997)] proposed a numerical algorithm which gives a constructive proof of the theorem. \textit{M. Köppen} [Lect. Notes Comput. Sci. 2415, 474--479 (2002; Zbl 1013.68811)] observed that the Sprecher argumentation contains some gaps and suggested modifications, but without any proof. In this paper the authors, using the idea of Köppen, show that Sprecher's algorithm converges. It seems, it is the first correct constructive proof of Kolmogorov's superposition theorem.
Mathematics Subject Classification ID: 26B40
zbMATH DE Number: 5659502
zbMATH Keywords: Komogorov's superposition theorem
zbMATH Keywords: superposition of functions
reviewed by: Andrzej Nowak
MaRDI profile type: Publication
full work available at URL: s00365-009-9054-2
OpenAlex ID: W2062368570
cites work: Implications and applications of Kolmogorov's superposition theorem
cites work: Q5794159
cites work: Q5691132
cites work: Q3263816
cites work: Q3265072
cites work: Q4706528
cites work: Q5534403
cites work: Q4851733
cites work: Approximative versions of Kolmogorov's superposition theorem, proved constructively
cites work: Dimension of metric spaces and Hilbert’s problem 13
cites work: On the Structure of Continuous Functions of Several Variables
cites work: An improvement in the superposition theorem of Kolmogorov
DOI: 10.1007/S00365-009-9054-2
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description: scientific article; zbMATH DE number 5659502