radialbasis

Radial basis functions, often abbreviated as RBFs, are real-valued functions whose value depends only on the distance from some origin point. In other words, for a function $ \phi $, $ \phi(x) = \phi(||x - c||) $, where $ c $ is the center of the function and $ ||.|| $ denotes a norm. Commonly, the Euclidean norm is used. The simplest example of a radial basis function is the Gaussian function, $ \phi(r) = e^{-(\epsilon r)^2} $, where $ \epsilon $ is a shape parameter.

Radial basis functions are widely used in interpolation, approximation, and machine learning. In interpolation, a function is approximated by a linear combination of RBFs centered at data points. This approach is particularly useful for scattered data, where standard polynomial interpolation can be problematic. The approximation is typically of the form $ f(x) = \sum_{i=1}^N w_i \phi(||x - c_i||) $, where $ w_i $ are weights and $ c_i $ are the centers.

The choice of the RBF kernel and its parameters significantly impacts the performance of the approximation. Different kernels, such as the multiquadric, inverse multiquadric, and thin-plate spline, offer varying properties and are suited to different types of data and problems. RBFs are also employed in solving partial differential equations and in areas like computer graphics for surface reconstruction and animation. Their ability to efficiently handle high-dimensional data and their relatively simple mathematical structure make them a versatile tool in scientific computing and data analysis.