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For the polynomial kernel, $K(x,y) = (x^Ty+c)^d$, the implicit feature space $\phi$ for which $K(x,y) = \phi(x)^T \phi(y)$ is of finite dimension and well known [1][2].

It is also well known that the implicit feature space of gaussian kernel $K(x,y) = e^{-||x-y||^2/2\sigma^2}$ is of infinite dimension [2].

My question is, what is the implicit feature space of power kernel?: $K(x,y) = ||x-y||^d$

What I have so far (may be wrong):

For $d = 2$ and $x,y \in ℝ^2$, $K(x,y) = ||x-y||^2 = (x_1 - y_1)^2 + (x_2 - y_2)^2 = x_1^2 + y_1^2 -2x_1y_1 + x_2^2 + y_2^2 -2x_2y_2$

but I cant find any definition of $\phi(x) $ so that $K(x,y) = \phi(x)^T \phi(y)$

thanks in advance.

References

[1] https://en.wikipedia.org/wiki/Polynomial_kernel

[2] http://arxiv.org/pdf/0904.3664v1.pdf pags. 37-39

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