RadialbasisFunktionsInterpolation

RadialbasisFun is a term that can refer to a few related concepts in mathematics and computer science, primarily concerning radial basis functions (RBFs). Radial basis functions are real-valued functions whose value depends only on the distance from some origin point. This means that for a function $f$ and a center point $c$, $f(x) = \phi(||x - c||)$, where $|| \cdot ||$ denotes a norm, typically the Euclidean norm, and $\phi$ is a univariate function. The shape of the function $\phi$ determines the specific type of RBF. Common examples include the multiquadric, the inverse multiquadric, the Gaussian, and the thin plate spline.

RBFs are widely used in function approximation, interpolation, and machine learning. In function approximation, a desired function is represented as a linear combination of RBFs centered at a set of data points. This approach is particularly effective for interpolating scattered data, where the data points are not arranged in a regular grid. The centers of the RBFs are often chosen to be the data points themselves.

In machine learning, RBFs are famously used as activation functions in neural networks, specifically in radial basis function networks (RBFNs). These networks have a hidden layer composed of RBF units, and their output layer is typically a linear combination of the hidden layer's outputs. RBF kernels are also fundamental to support vector machines (SVMs), providing a way to implicitly map data into a higher-dimensional feature space where it may be linearly separable. The choice of RBF and its parameters (like the width or shape parameter) significantly impacts the performance of these models.