KernelFamilien

KernelFamilien is a term used in machine learning to refer to families of kernel functions that define similarities between data points without explicit feature maps. A kernel function k(x, y) belongs to a kernel family if it is symmetric and positive semi-definite, properties that guarantee the existence of a feature map phi into a reproducing kernel Hilbert space (RKHS) such that k(x, y) = <phi(x), phi(y)>.

According to Mercer's theorem, such kernels induce valid inner products in potentially infinite-dimensional spaces, enabling the kernel trick to apply linear methods in nonlinear feature spaces.

Common kernel families include Gaussian/Radial Basis Function (k = exp(-||x - y||^2 / (2 l^2))), Polynomial (k = (alpha x^T y + coef0)^degree), Linear (k = x^T y), Laplacian (k = exp(-||x - y||_1 / l)), and Matérn variants. Each family uses hyperparameters (length-scale, degree, coef0) that control smoothness and mapping dimension. Users often combine kernels via sums and products to create more flexible kernels. Hyperparameters are tuned by cross-validation or marginal likelihood in Gaussian processes.

KernelFamilien are used in a range of kernel-based methods, including support vector machines, kernel ridge regression, Gaussian process regression, spectral clustering, and kernel k-means. Kernel selection affects bias-variance and can be sensitive to data; scalability and numerical stability are considerations.