GaussiskRBF

GaussiskRBF, or the Gaussian radial basis function, is a radial basis function based on the Gaussian distribution. It is commonly defined for inputs x and x' in Euclidean space as phi(x, x') = exp(-||x - x'||^2 / (2 sigma^2)), or equivalently phi(r) = exp(-gamma r^2) with gamma = 1/(2 sigma^2). The function is symmetric, infinitely differentiable, and strictly positive for all distinct inputs. As a kernel it is stationary and isotropic, depending only on the distance between inputs. In machine learning, GaussiskRBF is widely used as a kernel in methods such as support vector machines, Gaussian process regression, and kernel ridge regression, and as the basis functions in radial basis function networks. The kernel trick allows these algorithms to operate in high-dimensional feature spaces without explicit mappings; the resulting Gram matrix has entries K_ij = phi(x_i, x_j).

Hyperparameters sigma or gamma control the kernel’s width and thus the locality of influence. A large gamma (small sigma) yields a narrow, highly localized kernel that fits data closely but risks overfitting; a small gamma (large sigma) produces a smoother, more global influence that may underfit. Selecting these parameters typically involves cross-validation, grid search, or Bayesian optimization. In radial basis function networks, centers and widths of the Gaussian basis functions determine the fit and are often learned from data.

GaussiskRBF is a positive definite kernel with smooth, flexible behavior, making it a popular choice for regression, classification, and interpolation tasks. Its performance depends on appropriate tuning of the width parameter and, for large datasets, on scalable approximation techniques.