rbf

RBF stands for radial basis function, a real-valued function whose value depends on the distance from a center, typically through a radial argument φ(||x − c||). In machine learning and approximation theory, radial basis functions are used as elementary building blocks for interpolation, smoothing, and function approximation. A common and widely used shape is the Gaussian: φ(r) = exp(−r^2/(2σ^2)). Other forms include multiquadric, inverse quadratic, and thin-plate spline.

In a radial basis function network, a hidden layer consists of units that compute φ(||x − c_i||) for a set of centers c_i, followed by a linear output layer. The network output is y(x) = Σ_i w_i φ(||x − c_i||) + b. Centers can be chosen by clustering (for example, k-means), grid sampling, or learned jointly; widths can be fixed or adapted. With sufficient units and appropriately chosen parameters, RBF networks can approximate a wide range of continuous functions.

In kernel methods, the RBF kernel is a function k(x, x') = exp(−γ ||x − x'||^2), used in support vector machines, kernel ridge regression, and Gaussian processes. This kernel implicitly maps inputs into an infinite-dimensional feature space, enabling flexible, nonlinear learning while preserving certain mathematical properties. The Gaussian kernel is valued for locality and smoothness, and it underpins connections to Gaussian process modeling via the squared-exponential covariance.

The acronym RBF also has other meanings outside mathematics and computer science, such as resting bitch face; this article focuses on radial basis functions.