relations by applying a scaling operation from ${\mathbb{R}}$ to $[0,1]$.

Ordinal-valued relations: situated somewhat in the middle between the other two settings, here it is assumed that the actual values of the relation do not matter but rather the provided order information should be learned.

Furthermore, one can integrate different types of domain knowledge in our framework, by guaranteeing that certain properties are satisfied. The following cases can be distinguished:

Symmetric relations. Applications arise in many domains and metric learning or learning similarity measures can be seen as special cases that require additional properties to hold, such as the triangle inequality for metrics and positive definiteness or transitivity properties for similarity measures. As shown below, learning symmetric relations can be interpreted as learning edges in an undirected graph.

Reciprocal or antisymmetric relations. Applications arise here in domains such as preference learning, game theory and bioinformatics for representing preference relations, choice probabilities, winning probabilities, gene regulation, etc. We will provide a formal definition below, but, given a rescaling operation from ${\mathbb{R}}$ to $[0,1]$, antisymmetric relations can be converted into reciprocal relations. Similar to symmetric relations, transitivity properties typically guarantee additional constraints that are definitely required for certain applications. It is, for 