5
$\begingroup$

Consider the polynomial kernel: $$K(\boldsymbol{x}, \boldsymbol{x}') = (\boldsymbol{x}^{T} \boldsymbol{x}'+c)^{d}$$

This kernel satisfies the Mercer's theorem/condition. Since I never saw any restriction on the degree d I assumed it would also be possible to use a degree in the range ]0..1]. However when I computed the Eigenvalues of my corresponding kernel matrix it turned out that not all of them were positive which is a requirement for a positive semi-definite kernel function. I now believe that the degree has to be greater than 1 but all scientific papers I read about polynomial kernels make no restriction regarding this parameter. Can you point me to a paper where they state that restriction or am I just wrong in my assumption?

$\endgroup$

1 Answer 1

4
$\begingroup$

Check page 89 of GPML , available for download as PDF. You were right.

$\endgroup$
1
  • 2
    $\begingroup$ thanks for this references. Now I've even found a paper where this is proved (p.5 example 1) $\endgroup$
    – nico1510
    Jan 14, 2015 at 20:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

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