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

作者：Tikk D^*; Koczy LT; Gedeon TD

来源：International Journal of Approximate Reasoning, 2003, 33(2): 185-202.

DOI：10.1016/S0888-613X(03)00021-5

• universal approximation performed by fuzzy systems and neural networks
• Kolmogorov's theorem
• approximation behaviour of soft computing techniques
• course of dimensionality
• nowhere denseness
• approximation rates
• constructive proofs

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 epsilon 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 epsilon 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.

• 出版日期2003-6