Effects of moving the centers in an RBF network

Abstract—In radial basis function (RBF) networks, placement of centers is said to have a significant effect on the performance of the network. Supervised learning of center locations in some ap-plications show that they are superior to the networks whose cen-ters are located using unsupervised methods. But such networks can take the same training time as that of sigmoid networks. The increased time needed for supervised learning offsets the training time of regular RBF networks. One way to overcome this may be to train the network with a set of centers selected by unsupervised methods and then to fine tune the locations of centers. This can be done by first evaluating whether moving the centers would de-crease the error and then, depending on the required level of ac-curacy, changing the center locations. This paper provides new results on bounds for the gradient and Hessian of the error con-sidered first as a function of the independent set of parameters, namely the centers, widths, and weights; and then as a function of centers and widths where the linear weights are now functions of the basis function parameters for networks of fixed size. Moreover, bounds for the Hessian are also provided along a line beginning at the initial set of parameters. Using these bounds, it is possible to es-timate how much one can reduce the error by changing the centers. Further to that, a step size can be specified to achieve a guaranteed amount of reduction in error. Index Terms—Generalized methods, gradient methods, Hessian matrices, intelligent networks, learning systems, neural-networ