  Recent years have witnessed the successful application of low-dimensional
vector space representations of knowledge graphs to predict missing facts or
find erroneous ones. However, it is not yet well-understood to what extent
ontological knowledge, e.g. given as a set of (existential) rules, can be
embedded in a principled way. To address this shortcoming, in this paper we
introduce a general framework based on a view of relations as regions, which
allows us to study the compatibility between ontological knowledge and
different types of vector space embeddings. Our technical contribution is
two-fold. First, we show that some of the most popular existing embedding
methods are not capable of modelling even very simple types of rules, which in
particular also means that they are not able to learn the type of dependencies
captured by such rules. Second, we study a model in which relations are
modelled as convex regions. We show particular that ontologies which are
expressed using so-called quasi-chained existential rules can be exactly
represented using convex regions, such that any set of facts which is induced
using that vector space embedding is logically consistent and deductively
closed with respect to the input ontology.
