Positive-semidefinite kernel

A symmetric function $k$ is a positive-semidefinite kernel when every finite Gram matrix $(k(x_i,x_j))$ is positive semidefinite.

For points $x_1,\dots ,x_n$, the kernel matrix has entries $K_{ij}=k(x_i,x_j)$ and is positive semidefinite.

A reproducing-kernel Hilbert space is a Hilbert space of functions in which point evaluation is continuous and satisfies $f(x)=⟨f,k(x,\cdot )⟩$.

The representer theorem reduces regularized optimization over a reproducing-kernel Hilbert space to the span of kernel sections at the observed points.

Every positive-semidefinite kernel $k$ has a Hilbert-space feature map $\varphi$ satisfying $k(x,y)=⟨\varphi (x),\varphi (y)⟩$, and determines a unique reproducing-kernel Hilbert space.

For feature vectors $\varphi (x_i)$, the empirical uncentered kernel covariance operator is

The kernel trick evaluates feature-space inner products through $k(x,y)$ without explicitly constructing the feature vectors.

Kernel principal component analysis diagonalizes a centered kernel matrix to perform principal component analysis in an implicit feature space.

If a stationary Gaussian process has covariance spectral density $\hat{K}$, its reproducing-kernel Hilbert space has normon the functions for which this integral is finite.