  Hierarchical relations are prevalent and indispensable for organizing human
knowledge captured by a knowledge graph (KG). The key property of hierarchical
relations is that they induce a partial ordering over the entities, which needs
to be modeled in order to allow for hierarchical reasoning. However, current KG
embeddings can model only a single global hierarchy (single global partial
ordering) and fail to model multiple heterogeneous hierarchies that exist in a
single KG. Here we present ConE (Cone Embedding), a KG embedding model that is
able to simultaneously model multiple hierarchical as well as non-hierarchical
relations in a knowledge graph. ConE embeds entities into hyperbolic cones and
models relations as transformations between the cones. In particular, ConE uses
cone containment constraints in different subspaces of the hyperbolic embedding
space to capture multiple heterogeneous hierarchies. Experiments on standard
knowledge graph benchmarks show that ConE obtains state-of-the-art performance
on hierarchical reasoning tasks as well as knowledge graph completion task on
hierarchical graphs. In particular, our approach yields new state-of-the-art
Hits@1 of 45.3% on WN18RR and 16.1% on DDB14 (0.231 MRR). As for hierarchical
reasoning task, our approach outperforms previous best results by an average of
20% across the three datasets.
