Local Modifications to Radial Basis Networks

Abstract

The form of a radial basis network is a linear combination of translates of a given radial basis function, φ(r). The radial basis method involves determining the values of the unknown parameters within the network given a set of inputs, {x _k }, and their corresponding outputs, {f _k }. It is usual for some of the parameters of the network to be fixed. If the positions of the centres of the basis functions are known and constant, the radial basis problem reduces to a standard linear system of equations and many techniques are available for calculating the values of the unknown coefficients efficiently. However, if both the positions of the centres and the values of the coefficients are allowed to vary, the problem becomes considerably more difficult. A highly non-linear problem is produced and solved in an iterative manner. An initial guess for the best positions of the centres is made and the coefficients for this particular choice of centres are calculated as before. For each iteration, a small change to the position of the centres is made in order to improve the quality of the network and the values of the coefficients for these new centre positions are determined. The overall algorithm is computationally expensive and here we consider ways of improving the efficiency of the method by exploiting the local stability of the thin plate spline basis function. At each step of the iteration, only a small change is made to the positions of the centres and so we can reasonably expect that there is only a small change to the values of the corresponding coefficients. These small changes are estimated using local modifications.