For the two-dimensional case, where $\boldsymbol X=[x_1, x_2]$ and its corresponding expanded represetation $\boldsymbol\phi(X)= [1, \sqrt2 x_1, \sqrt2 x_2, x_1^2, x_2^2, \sqrt2x_1x_2]$, we can derivate the polynomial function kernel $K(X,X^T)=<\phi(X),\phi(X)^T>=(1+<X,X^T>)^2 $.

What is the expanded representation, $\boldsymbol \phi(X)$, required to obtaing the RBF kernel $ K(X,X^T)=<\phi(X),\phi(X)^T>=exp(-\sigma||X-X^T||^2)$?


The feature space of the RBF Kernel actually has infinite dimension, so you cannot write it in the form of a fully expanded representation. However, you can write it as a sum.

I will link to some course notes:

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Source: https://arxiv.org/pdf/0904.3664v1.pdf


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