For example, how to prove
$\forall(x,y)\in R^N\times R^N,K(x,y) = \displaystyle\frac{1}{1+\frac{||x - y||^2}{{\sigma}^2}}\\$
where $\sigma > 0$ is a parameter, is positive definite? I have tried to construct the target kernel base on some negative definite kernel, such as the square distance $||x-y||^2$, but doesn't seems to work.



How to prove that a kernel is positive definite?
Here is a simple trick that relies only on elementary ideas.


First, we use the observation that for $t>0$
\begin{equation\*}
 \exp(-t\|x-y\|^2) = \exp(-t\|x\|^2)\exp(2t\langle x, y\rangle)\exp(-t\|y\|^2),
\end{equation\*}
which is clearly positive definite (use $\langle x,y\rangle^k$ is pd for $k\ge0$).


Now, just use the integral
\begin{equation\*}
 \frac{1}{1+\|x-y\|^2} = \int\_0^\infty e^{-t(1+\|x-y\|^2)}dt,
\end{equation\*}
which is basically just a nonnegative sum of positive definite functions, hence itself positive definite.


