yields the Kronecker product edge kernel (a.k.a. the tensor product pairwise kernel) in the dual representation: $$\begin{aligned}
\label{eq:tppk}
{K}_{\otimes}^{\Phi}({e},{\overline{{e}}}) = {K}_{\otimes}^{\Phi}({v},{v}',{\overline{{v}}},{\overline{{v}}}')= {K}^{\phi}({v},{\overline{{v}}}) {K}^{\phi}({v}',{\overline{{v}}}') \,,\end{aligned}$$ with $K^{\phi}$ the kernel corresponding to $\phi$.

This section aims to formally prove that the Kronecker product edge kernel is the best kernel one can choose, when no further domain knowledge is provided about the underlying relation that generates the data. We claim that with an appropriate choice for $K^{\phi}$, such as the Gaussian RBF kernel, the kernel $K^{\Phi}$ generates a class ${\mathcal{H}}$ of universally approximating functions for learning any type of relation. Armed with the definition of universality for kernels and the Stone-Weierstra[ß]{} theorem [@Steinwart2002consistency], we arrive at the following theorem concerning the Kronecker product pairwise kernels:

\[unikrontheorem\] Let us assume that the space of nodes ${\mathcal{V}}$ is a compact metric space. If a continuous kernel ${K}^\phi$ is universal on ${\mathcal{V}}$, then ${K}_{\otimes}^\Phi$ defines a universal kernel on ${\mathcal{E}}$.

The proof can be found in the appendix. We would like to emphasize that one cannot conclude from the theorem that the Kronecker product pairwise kernel is the best kernel to use in all possible situations. The theorem only shows that the Kronecker product pairwise kernel 