A survey on universal approximation and its limits in soft computing techniques

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Abstract

This paper deals with the approximation behaviour of soft computing techniques. First, we give a survey of the results of universal approximation theorems achieved so far in various soft computing areas, mainly in fuzzy control and neural networks. We point out that these techniques have common approximation behaviour in the sense that an arbitrary function of a certain set of functions (usually the set of continuous function, C) can be approximated with arbitrary accuracy ε on a compact domain. The drawback of these results is that one needs unbounded numbers of "building blocks" (i.e. fuzzy sets or hidden neurons) to achieve the prescribed ε accuracy. If the number of building blocks is restricted, it is proved for some fuzzy systems that the universal approximation property is lost, moreover, the set of controllers with bounded number of rules is nowhere dense in the set of continuous functions. Therefore it is reasonable to make a trade-off between accuracy and the number of the building blocks, by determining the functional relationship between them. We survey this topic by showing the results achieved so far, and its inherent limitations. We point out that approximation rates, or constructive proofs can only be given if some characteristic of smoothness is known about the approximated function.

Original languageEnglish
Pages (from-to)185-202
Number of pages18
JournalInternational Journal of Approximate Reasoning
Volume33
Issue number2
DOIs
Publication statusPublished - Jun 2003

Keywords

  • Approximation behaviour of soft computing techniques
  • Approximation rates
  • Constructive proofs
  • Course of dimensionality
  • Kolmogorov's theorem
  • Nowhere denseness
  • Universal approximation performed by fuzzy systems and neural networks

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Artificial Intelligence
  • Applied Mathematics

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