The Kernel Trick is a technique in machine learning that allows one to write a certain type of bivariate function $$k : \mathcal{X}\times\mathcal{X}\to\mathbb{R}$$ as an inner product on some Hilbert space $\mathcal{V}$. This is done via what is known as a feature map $\varphi: \mathcal{X}\to\mathcal{V}$, which satisfies
$$k(x,x') = \langle \varphi(x), \varphi(x')\rangle_{\mathcal{V}}.$$
This is to say one can represent a (potentially non-linear) kernel via the composition of a
- potentially non-linear feature map $\varphi$, and
- an inner product.
Is there some explicit list of known kernel maps + feature map pairs? In briefly reviewing materials about the kernel trick, it appears that in machine learning practitioners often do not care about the specifics of $\varphi(x)$. I do though, as in my application it appears useful to know things like the gradient/Lipschitz constant of $\varphi$.