Gaussian kernel ($k(x,y)=e^{-\Vert x-y{\Vert }^2/(2{\sigma }^2)}$)

For $\sigma >0$, the Gaussian kernel on a Euclidean space isIt is a translation-invariant positive-semidefinite kernel, with $k(x,x)=1$. The scale $\sigma$ controls the distance over which comparisons remain large: $\Vert x-y\Vert =\sigma$ gives $k(x,y)=e^{-1/2}$. Its feature map requires an infinite-dimensional vector space on the full Euclidean space. Nonlinearity therefore enters through the representation even when a later method is linear in those features.