On a Constructive Proof of Kolmogorov's Superposition Theorem
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| Title: | Main Title: On a Constructive Proof of Kolmogorov's Superposition Theorem |
| Description: | 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: | 10.1007/s00365-009-9054-2 (DOI) |
Responsible Party
| Creators: | Andreas C. Braun (Author), Michael Griebel (Author) |
| Publisher: | Springer |
| Publication Year: | 2013 |
Topic
| TR32 Topic: | Other |
| Related Subproject: | D5 |
| Subjects: | Keywords: Kolmogorov’s Superposition Theorem, Superposition of Functions, Representation of Functions |
File Details
| Filename: | 2009_Braun_CA.pdf |
| Data Type: | Text - Article |
| Size: | 23 Pages |
| File Size: | 626 KB |
| Dates: | Accepted: 11.11.2008 Issued: 16.05.2009 |
| Mime Type: | application/pdf |
| Data Format: | |
| Language: | English |
| Status: | Completed |
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| Download Permission: | Only Project Members |
| General Access and Use Conditions: | For internal use only |
| Access Limitations: | For internal use only |
| Licence: | [TR32DB] Data policy agreement |
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Specific Information - Publication
| Publication Status: | Published |
| Review Status: | Peer reviewed |
| Publication Type: | Article |
| Article Type: | Journal |
| Source: | Constructive Approximation |
| Issue: | 3 |
| Volume: | 30 |
| Number of Pages: | 23 (653 - 675) |
Metadata Details
| Metadata Creator: | Harrie-Jan Hendricks-Franssen |
| Metadata Created: | 03.12.2013 |
| Metadata Last Updated: | 03.12.2013 |
| Subproject: | D5 |
| Funding Phase: | 1 |
| Metadata Language: | English |
| Metadata Version: | V50 |
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| Metadata Downloads: | 0 |
| Dataset Downloads: | 0 |
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