= Gaussian kernel
{c}
{title2=$k(x,y)=e^{-\lVert x-y\rVert^2/(2\sigma^2)}$}
For $\sigma>0$, the <Gaussian kernel> on a <Euclidean space> is
$$
k(x,y)=\exp\!\left(-\frac{\|x-y\|^2}{2\sigma^2}\right).
$$
It is a translation-invariant <positive-semidefinite kernel>, with $k(x,x)=1$. The scale $\sigma$ controls the distance over which comparisons remain large: $\|x-y\|=\sigma$ 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.
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