Exact trade-off between approximation accuracy and interpretability: solving the saturation problem for certain FRBSs

Although, in literature various results can be found claiming that fuzzy rule-based systems (FRBSs) possess the universal approximation property, to reach arbitrary accuracy the necessary number of rules are unbounded. Therefore, the inherent property of FRBSs in the original sense of Zadeh, namely that they can be characterized by a semantic relying on linguistic terms is lost. If we restrict the number of rules, universal approximation is not valid anymore as it was shown for, including others, Sugeno and TSK type models [10,19]. Due to this theoretic bound there is recently a great demand among researchers on finding trade-off techniques between a required accuracy and the number of rules, and as such, they attempt to determine the (optimal) number of rules as a function of accuracy. Naturally, to obtain such results one has to restrict somehow the set of continuous functions, usually requiring some smoothness conditions on the approximated function. In terms of approximation theory this is the so-called saturation problem, the determination of optimal order and class of approximation. Hitherto, saturation classes and orders have not been determined for FRBSs and neural networks. In this paper we solve the saturation problem for a special type of fuzzy controller, for the generalized KH-interpolator, being a suitable inference method in sparse rule bases.