How to understand mapping function of kernel?

Question

For a kernel function, we have two conditions one is that it should be symmetric which is easy to understand intuitively because dot products are symmetric as well and our kernel should also follow this. The other condition is given below

There exists a map $φ:R^d→H$ called kernel feature map into some high dimensional feature space H such that $∀x,x'$ in $R^d:k(x,x') = <φ(x),φ(x')>$

I understand that this means that there should exist a feature map that will project the data from low dimension to any high dimension $D$ and kernel function will take the dot product in that space.

For example, the Euclidean distance is given as

$d(x,y)=∑_i(x_i−y_i)^2=<x,x>+<y,y>−2<x,y>$

If I look this in terms of second condition how do we know that doesn't exist any feature map for euclidean distance? What exactly are we looking in feature maps mathematically?

Answer 1

A kernel function $f : \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$ is a valid support vector kernel if it is a Mercer kernel. Mercer's condition essentially ensures that the Gram matrix of the kernel is positive semi-definite. Interestingly, this ensures that the SVM objective is convex.

The Euclidean distance function does not satisfy Mercer's condition since it's Gram matrix is not necessary positive semi-definite. Thus, it is not a valid kernel.