RBF Neural Network: A Three-Layer Feedforward Network with Single Hidden Layer

The Radial Basis Function (RBF) neural network is a structurally compact yet powerful feedforward neural network architecture. Its typical configuration consists of three layers: input layer, hidden layer, and output layer, where the hidden layer employs radial basis functions as activation functions.

The core characteristic of this network lies in its hidden layer neurons using radially symmetric basis functions (such as Gaussian functions). When input signals approach the center of these basis functions, the hidden layer neurons generate significant output. This localized response property makes RBF networks particularly effective for function approximation and pattern recognition tasks. In code implementation, the Gaussian function typically appears as: φ(x) = exp(-||x - c||² / (2σ²)), where c represents the center and σ controls the width of the radial basis function.

During the training process, three key parameters must be determined: basis function centers, width parameters, and output layer connection weights. Common training strategies involve: first determining basis function centers through clustering methods (like k-means algorithm), then computing width parameters (often based on inter-center distances), and finally solving output layer weights using least squares methods. The k-means clustering algorithm implementation typically involves iterative centroid updates, while the weight calculation can be efficiently solved using matrix operations: W = (ΦᵀΦ)⁻¹ΦᵀY, where Φ is the hidden layer output matrix and Y contains target values.

Compared to traditional multi-layer perceptrons, RBF neural networks offer advantages such as faster training speeds and reduced susceptibility to local minima. These characteristics make them particularly suitable for solving nonlinear classification problems, function approximation tasks, and time series prediction applications. The network's efficiency stems from its linear output layer, which allows for direct analytical solutions rather than iterative gradient-based optimization.