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A systematic four-step batch approach is presented for the second-order training of radial basis function (RBF) neural networks for estimation. First, it is shown that second-order training works best when applied separately to several disjoint parameter subsets. Newton's method is used to find distance measure weights, leading to a kind of embedded feature selection. Next, separate Newton's algorithms are developed for RBF spread parameters, center vectors, and output weights. The final algorithm's training error per iteration and per multiply are compared to those of other algorithms, showing that convergence speed is reasonable. For several widely available datasets, it is shown that tenfold testing errors of the final algorithm are less than those for recursive least squares, the error correction algorithm, the support vector regression training, and Levenberg-Marquardt.

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In this paper, we proposed an hybrid optimal radial-basis function (RBF) neural network for approximation and illumination invariant image segmentation. Unlike other RBF learning algorithms, the proposed paradigm introduces a new way to train RBF models by using optimal learning factors (OLFs) to train the network parameters, i.e. spread parameter, kernel vector and a weighted distance measure (DM) factor to calculate the activation function. An efficient second order Newton's algorithm is proposed for obtaining multiple OLF's (MOLF) for the network parameters. The weights connected to the output layer are trained by a supervised-learning algorithm based on orthogonal least square (OLS). The error obtained is then back-propagated to tune the RBF parameters. By applying RBF network for approximation on some real-life datasets and classification to reduce illumination effects of image segmentation, the results show that the proposed RBF neural network has fast convergence rates combining with low computational time cost, allowing it a good choice for real-life application such as image segmentation.

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