Adaptive thinning of centers for approximation of a large data set by radial functions

Abstract

Given data at a large set of centers, the problem investigated in this article, is how to choose a smaller subset of centers such that the least squares radial function approximation will be with predefined accuracy. Our solution is based on an adaptive thinning strategy, removing in a greedy way less significant centers one by one so as to minimize an anticipated error. The novelty in our approach is the replacement of the anticipated error by simpler “predicting” functionals. A predicting functional is a functional which chooses with a high probability the same centers to be removed as an anticipated error. We derive several such predicting functionals for specific radial functions which have the “functionals consistency” property, namely that there are functionals which change their values on translates of the radial function in a co-monotone way. Our numerical tests demonstrate good performance of the proposed predicting functionals on piecewise continuous functions.