kernelfunktio

A kernelfunktion, or kernel function, is a function k: X × X → R that measures the similarity between two inputs. In many settings it corresponds to an inner product in a (potentially high‑dimensional) feature space: k(x, y) = φ(x) · φ(y). This interpretation allows nonlinear learning algorithms to operate in implicit high-dimensional spaces by computing kernel evaluations directly in the input space.

Key properties of a kernel include symmetry and positive semidefiniteness. A kernel is symmetric if k(x, y) = k(y, x) for all x and y, and it is positive semidefinite if, for any finite set {x_i}, the Gram matrix K with entries K_ij = k(x_i, x_j) is positive semidefinite. Mercer’s theorem provides conditions under which such kernels correspond to inner products in some feature space.

Common examples of kernels are the linear kernel k(x, y) = x · y, the polynomial kernel (x · y + c)^d, the Gaussian or radial basis function (RBF) kernel exp(-||x − y||^2 / (2σ^2)), the Laplacian kernel exp(-||x − y|| / σ), and occasionally sigmoid-type kernels.

Applications span machine learning and statistics. Kernels enable the kernel trick for algorithms such as support vector machines, kernel principal component analysis, and kernel ridge regression. They are also used in kernel density estimation to smooth data and, in analysis, to define integral operators (Kf)(x) = ∫ k(x, y) f(y) dy and study their spectral properties.

Computationally, kernel methods require forming and manipulating an n × n Gram matrix, with costs growing at least quadratically with the number of samples. For large datasets, approximations such as Nyström methods or random Fourier features are commonly employed to reduce complexity.