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I understand that each kernel implies a particular feature map. For instance for $x,z \in R^2$ the kernel $K(x,z)=(dot(x,z))^2$ implies a feature map $φ(<x_1, x_2>)=<x_1^2 , x_1 x_2 , x_1 x_2, x_2^2>$, as though my inputs are transformed from $R^2$ to $R^4$ in the process of computing $K(x,z)$.

The polynomial kernels are clear enough to see in this way. However the radial basis function kernel in particular, i.e. $K(x,z) = exp(-||x-z||^2/c^2)$, supposedly implies an infinite dimensional feature map. Exactly why is this so?

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Try expanding the exponential into an infinite series, and you'll see it. Abu-Mostafa has a great demonstration of this, in his classic style.

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