Why clustering in function approximation? Theoretical explanation

Function approximation is a very important practical problem: in many pract ical applications, we know the exact form of the functional dependence y = f(x(1),..., x(n)) between physical quantities, but this exact dependence is complicated, so we need a lot of computer space to store it, and a lot of time to process it, i.e., to predict y from the given xi. It is therefore n ecessary to find a simpler approximate expression g(x(1),..., x(n)) approxi mate to f(x(1),..., x(n)) for this same dependence. This problem has been a nalyzed in numerical mathematics for several centuries, and it is, therefor e, one of the most thoroughly analyzed problems of applied mathematics. The re are many results related to approximation by polynomials, trigonometric polynomials, splines of different type, etc. Since this problem has been an alyzed for so long, no wonder that for many reasonable formulations of the optimality criteria, the corresponding problems of finding the optimal appr oximations have already been solved. Lately, however, new clustering-relate d techniques have been applied to solve this problem (by Yager, Filev, Chu, and others). At first glance, since for most traditional optimality criter ia, optimal approximations are already known, the clustering approach can o nly lead to non-optimal approximations, i.e., approximations of inferior qu ality. We show, however, that there exist new reasonable criteria with resp ect to which clustering-based function approximation is indeed the optimal method of function approximation. (C) 2000 John Wiley & Sons, Inc.