A variable projection method for the general radial basis function neural network

The variable projection (VP) method is a classical and effective method for the separable nonlinear least squares (SNLLS) problem. Training a radial basis function neural network (RBFNN) with only one output neuron by minimizing the sum of the squared errors (SSE) is an SNLLS problem, so that the classical VP method has been applied to RBFNN. However, the one-output-RBFNN (ORBFNN) is just one type of RBFNN, so that the paper proposes a new VP method for the general radial basis function neural network (GRBFNN) which has no limit of the number of the output neurons. The new VP method translates the problem corresponding to minimizing the SSE of GRBFNN into a lower-dimensional optimization problem. We prove theoretically that the set of stationary points of the objective function of the lower-dimensional problem is equivalent to that of the original objective function. In addition, the lower dimension leads to less guesses about the initial point for the new problem. The numerical experiments indicate that, with the same algorithm, minimizing the new objective function converges in fewer iterations and makes both a smaller training error and a testing error than minimizing the original objective function.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

The variable projection (VP) method is a classical and effective approach for solving the separable nonlinear least squares (SNLLS) problem. While the classical VP method has been applied to one-output radial basis function neural networks (ORBFNN), this study proposes a new VP method for general radial basis function neural networks (GRBFNN) that can have multiple output neurons. The new VP method transforms the SSE minimization problem of GRBFNN into a lower-dimensional optimization problem, and theoretical analysis shows that the stationary points of the lower-dimensional problem are equivalent to those of the original objective function. Numerical experiments demonstrate that minimizing the new objective function leads to faster convergence, smaller training errors, and smaller testing errors compared to minimizing the original objective function.