RBF Networks for Function Approximation

Radial Basis Function (RBF) Networks are neural network models specialized for function approximation, widely applied in pattern recognition, time series prediction, and nonlinear function approximation due to their simple architecture and strong approximation capabilities.

The core concept of RBF networks involves using radial basis functions (e.g., Gaussian functions) as activation functions for hidden layer neurons. By adjusting the centers and widths of these basis functions along with output layer weights, the network approximates target functions. Compared to Multi-Layer Perceptrons (MLP), RBF networks typically offer faster training speeds and superior local approximation performance.

Key algorithms for function approximation with RBF networks include:

Center Selection Algorithms: Determining optimal positions for hidden neuron centers using methods like K-means clustering or Orthogonal Least Squares (OLS). In code implementation, K-means clustering can be executed through iterative centroid updates while OLS employs Gram-Schmidt orthogonalization for sequential center selection.

Width Parameter Optimization: The Gaussian function's width parameter (σ) critically impacts generalization capability. Implementation approaches include cross-validation techniques or heuristic rules like the nearest neighbor distance method, where σ is set proportional to the average distance between centers.

Weight Learning: Output layer weights are optimized using Least Mean Squares (LMS) or gradient descent algorithms. The LMS solution can be computed directly through pseudoinverse operations (pinv() in MATLAB), while gradient descent requires iterative weight updates based on error backpropagation.

Compared to globally approximating networks like MLP, RBF networks demonstrate superior performance for functions with significant local variations, offering advantages in both computational efficiency and approximation accuracy.