Show that $\frac{1}{x+y}$ is a valid kernel

Question

Problem $5.2(c)$ in Foundations of Machine Learning (Mohri et al. 2012)

Show that $\kappa(x,y)=\frac{1}{x+y}$ is a valid PSD kernel over $(0, \infty) \times(0,\infty) $.

Answer 1

Note that $\kappa(x,y)=\int_0^\infty \exp(-tx)\exp(-ty)dt$, so $$\sum_{i,j} a_ia_j \kappa(x_i,x_j) = \int_0^\infty \left|\sum_i a_i\exp(-tx_i)\right|^2dt \ge0.$$

This gives you PSD, but you can also verify strict positive definiteness by noting that for no finite combination of $i_i$ is the sum $\sum_i a_i\exp(-tx_i)$ the zero function.