Kernelbased

Kernel-based methods (often written as kernel-based or kernel based) are a class of algorithms that rely on kernel functions to measure similarity between data points and to implicitly map data into high-dimensional feature spaces. Through the kernel trick, computations involving inner products in the feature space can be performed without explicit mapping, enabling nonlinear modeling with linear algorithms in the transformed space. Many kernel functions are positive semidefinite and correspond to inner products in a reproducing kernel Hilbert space (RKHS).

Common kernels include linear, polynomial, Gaussian or radial basis function (RBF), Laplacian, and sigmoid. The kernel trick allows algorithms such as support vector machines, kernel ridge regression, kernel logistic regression, and kernel principal component analysis to operate in rich feature spaces while retaining scalability compared to explicit feature mappings. Kernel density estimation uses kernels to estimate probability densities, and Gaussian processes use covariance kernels to define distributions over functions.

The theoretical foundation relies on Mercer's theorem, which characterizes valid kernels as reproducing inner products in an RKHS, and on the properties of positive semidefinite kernels. Some kernels provide universal approximation capabilities over certain input domains, enabling flexible modeling of complex patterns.

Practical considerations include selecting a kernel and tuning hyperparameters such as bandwidth or degree; cross-validation is commonly used. Computational costs typically scale with the number of samples, often O(n^2) time and O(n^2) memory for naïve implementations, leading to scalable variants like the Nyström method and random Fourier features. Kernel methods are widely used in classification, regression, density estimation, and dimensionality reduction across many domains.