[758] - On a Constructive Proof of Kolmogorov's Superposition Theorem

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Citation
Braun, A. C., Griebel, M., 2009. On a Constructive Proof of Kolmogorov's Superposition Theorem. Constructive Approximation, 30 (3), 653 - 675. DOI: 10.1007/s00365-009-9054-2.
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Identification
Title(s):Main Title: On a Constructive Proof of Kolmogorov's Superposition Theorem
Description(s):Abstract: Kolmogorov (Dokl. Akad. Nauk USSR, 14(5):953–956, 1957) showed that any multivariate continuous function can be represented as a superposition of one dimensional functions. The proof of this fact, however, was not constructive, and it was not clear how to choose the outer and inner functions Φq and ψq,p, respectively. Sprecher (Neural Netw. 9(5):765–772, 1996; Neural Netw. 10(3):447–457, 1997) gave a constructive proof of Kolmogorov’s superposition theorem in the form of a convergent algorithm which defines the inner functions explicitly via one inner function ψ by ψp,q := λpψ(xp+qa) with appropriate values λp, a ∈ R. Basic features of this function such as monotonicity and continuity were supposed to be true but were not explicitly proved and turned out to be not valid. Köppen (ICANN 2002, Lecture Notes in Computer Science, vol. 2415, pp. 474–479, 2002) suggested a corrected definition of the inner function ψ and claimed, without proof, its continuity and monotonicity. In this paper we now show that these properties indeed hold for Köppen’s ψ, and we present a correct constructive proof of Kolmogorov’s superposition theorem for continuous inner functions ψ similar to Sprecher’s approach.
Identifier(s):DOI: 10.1007/s00365-009-9054-2
Responsible Party
Creator(s):Author: Andreas C. Braun
Author: Michael Griebel
Publisher:Springer
Topic
TR32 Topic:Other
Subject(s):CRC/TR32 Keywords: Kolmogorov’s Superposition Theorem, Superposition of Functions, Representation of Functions
File Details
File Name:2009_Braun_CA.pdf
Data Type:Text
Size(s):23 Pages
File Size:626 kB (0.611 MB)
Date(s):Date Accepted: 2008-11-11
Issued: 2009-05-16
Mime Type:application/pdf
Data Format:PDF
Language:English
Status:Completed
Constraints
Download Permission:OnlyTR32
General Access and Use Conditions:For internal use only
Access Limitations:For internal use only
Licence:TR32DB Data policy agreement
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Measurement Region:None
Measurement Location:--None--
Specific Informations - Publication
Status:Published
Review:PeerReview
Year:2009
Type:Article
Article Type:Journal
Source:Constructive Approximation
Issue:3
Volume:30
Number Of Pages:23
Page Range:653 - 675
Metadata Details
Metadata Creator:Harrie-Jan Hendricks-Franssen
Metadata Created:2013-12-03
Metadata Last Updated:2013-12-03
Subproject:D5
Funding Phase:1
Metadata Language:English
Metadata Version:V40
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