easily incorporated in our framework.

Let $\Psi$ be a feature mapping on ${\mathcal{V}}^2$ and let ${h}$ be a hypothesis defined by (\[eq:primalmodel\]), then the relation $Q$ of type (\[eq:monmap\]) is reciprocal if $\Phi$ is given by $$\Phi_R({e})=\Phi_R({v},{v}')=\Psi({v},{v}')-\Psi({v}',{v}) \,.$$

The proof is immediate. In addition, one can easily show that reciprocity as domain knowledge can be enforced in the dual formulation. Let us in the least restrictive form now consider the Kronecker product for $\Psi$, then one obtains for $\Phi_R$ the kernel $K_{\otimes R}^{\Phi}$ given by $K_{\otimes R}^{\Phi}({e},{\overline{{e}}})=$ $$\begin{aligned}
\label{eq:recedgekernel}
2 \big(K^{\phi}({v},{\overline{{v}}}) K^{\phi}({v}',{\overline{{v}}}') - K^{\phi}({v},{\overline{{v}}}') K^{\phi}({v}',{\overline{{v}}})\big)\,.\end{aligned}$$ The following theorem shows that this kernel can represent any type of reciprocal relation.

\[antisymmetrictheorem\] Let $$R({\mathcal{V}}^2)=\{ t \mid t \in C({\mathcal{V}}^2),t({v},{v}')= - t({v}',{v})\}$$ be the space of all continuous antisymmetric relations from ${\mathcal{V}}^2$ to $\mathbb{R}$. If $K^\phi$ on ${\mathcal{V}}$ is universal, then for every function $t\in R({\mathcal{V}}^2)$ and every $\epsilon > 0$, there exists a function $h$ in the RKHS induced by the kernel $K_{\otimes R}^{\Phi}$ defined in (\[eq:recedgekernel\]), such that $$\label{recclaim}
\max_{({v},{v}')\in {\mathcal{V}}^2}\left\{\left\arrowvert t({v},{v}')-{h}({v},{v}')\right\arrowvert\right\}\leq\epsilon \,.$$

The proof can be found in the appendix.

Symmetric relations
-------------------

Symmetric relations form another important subclass of relations in our framework. As a specific type of symmetric 