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Is the left hand side of a feature map always $\phi(x)$ (for example, $\phi(x) = ...$)? Is it possible I have a feature map that's like $\phi (x^2) = x$ (If I already know the feature map is $\phi: \mathbb R^2 \rightarrow \mathbb R$? Also after I got the feature map, can I use K = $\phi(x)^T\phi(z)$ to get the kernel?

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The feature map is a function from your raw observations to what you put into a model, so unless you observe $x^2$ instead of $x$, yes, your function should act on $x$. In that case, though, I might call the observation $z$ and the mapped feature $\phi(z)=\sqrt{z}$.

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