Does a polynomial kernel with degree less than 1 satisfy Mercer's condition

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?

1 Answer

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

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