1
$\begingroup$

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)$?

$\endgroup$
2
$\begingroup$

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:

enter image description here

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.