we believe that situations where symmetric relations become Euclidean representable in a one-dimensional space occur very rarely, in contrast to reciprocal relations. The extension to $z > 1$ on the other hand does not guarantee the existence of one global ranking, then Euclidean representability still enforces some interesting properties, because it guarantees that the relation $Q$ is constructed from a Euclidean metric space with a dimension upper bounded by the number of nodes ${p}$. Moreover, this type of domain knowledge about relations can be incorporated in our framework. To this end, let $\Phi({v},{v}') = \Phi_S({v},{v}')$ and let $\Psi({v},{v}') = \phi({v}) \otimes (\phi({v}) - \phi({v}'))$ such that $K^{\Phi}$ becomes $$\begin{aligned}
\arraycolsep=2pt
K_{\rm MLPK}^{\Phi}({e},{\overline{{e}}}) &=& (K_{fR}^{\Phi}({e},{\overline{{e}}}))^2 \\
&=& \big(K^{\phi}({v},{\overline{{v}}}) + K^{\phi}({v}',{\overline{{v}}}') -K^{\phi}({v},{\overline{{v}}}') -
K^{\phi}({v}',{\overline{{v}}})\big)^2 \,.\end{aligned}$$ This kernel has been called the metric learning pairwise kernel by [@Vert2007]. As a consequence, the vector of parameters ${{\mathbf{w}}}$ can be rewritten as an ${r}\times {r}$ matrix ${\mathbf{W}}$ where ${\mathbf{W}}_{ij}$ corresponds to the parameter associated with $(\phi_i({v}) - \phi_i({v}'))(\phi_j({v}) - \phi_j({v}'))$ such that ${\mathbf{W}}_{ij} = {\mathbf{W}}_{ji}$.

\[thm:mlpk\] If ${\mathbf{W}}$ is positive semi-definite, then the symmetric relation $Q: {\mathcal{V}}^2 \rightarrow [0,1]$ given by (\[eq:monmap\]) with $h$ defined by (\[eq:primalmodel\]) and $K^{\Phi} = K_{\rm MLPK}^{\Phi}$ is an Euclidean 