to the representer theorem [@Scholkopf2002], any minimizer ${h}\in{\mathcal{H}}$ of (\[regalgorithm\]) admits a dual representation of the following form: $$\begin{aligned}
\label{eq:dualmodel}
{h}({\overline{{e}}})
= {{\mathbf{w}}}^T \Phi({\overline{{e}}}) = \sum_{{e}\in {\mathcal{E}}}
{a}_{{e}}{K}^{\Phi}({e},{\overline{{e}}}) \,,\end{aligned}$$ with ${a}_{{e}} \in\mathbb{R}$ dual parameters, ${K}^{\Phi}$ the kernel function associated with the RKHS and $\Phi$ the feature mapping corresponding to ${K}^{\Phi}$ and $$\begin{aligned}
{{\mathbf{w}}}=\sum_{{e}\in {\mathcal{E}}}
{a}_{{e}}\Phi({e}).\end{aligned}$$ We will alternate several times between the primal and dual representation for ${h}$ in the remainder of this article.

The primal representation as defined in (\[eq:primalmodel\]) and its dual equivalent (\[eq:dualmodel\]) yield an RKHS defined on edges in the graph. In addition, we will establish an RKHS defined on nodes, as every edge consists of a couple of nodes. Given an input space $\mathcal{V}$ and a kernel ${K}:\mathcal{V}\times\mathcal{V}\rightarrow\mathbb{R}$, the RKHS associated with ${K}$ can be considered as the completion of $$\left\{ f \in \mathbb{R}^\mathcal{V}  \left\arrowvert
f({v})=\sum_{i=1}^m\beta_i{K}({v},{v}_i)\right.\right\},$$ in the norm $$\Arrowvert f\Arrowvert_{K}=\sqrt{\sum_{i,j}\beta_i\beta_j{K}({v}_i,{v}_j)},$$ where $\beta_i\in\mathbb{R},m\in\mathbb{N},{v}_i\in\mathcal{V}$.

Learning arbitrary relations
----------------------------

As mentioned in the introduction, both crisp and graded relations can be handled by our framework. To make a subdivision between different cases, a loss function needs to be specified. For crisp relations, one can typically use the hinge loss, which is given by: $$\begin{aligned}
{\mathcal{L}}({h}({e}),{y}) = [1- 