In Gaussian Processes, SVMs, kernels are used (as to my understanding) as similarity measure. However, they have the constraint that any kernel has to be represented as a dot product. i.e. $k(x_1,x_2)=\langle f(x_1), f(x_2)\rangle$. Note that $f(x)$ could map x to a higher dimension e.g. $f(x)=[x^2\,\,\,\, \sqrt{2}x]^T$

My question is how can you derive the square exponential function $k_{SE}=\exp(-(x_1-x_2)^2)$ as a dot product.



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