DensE: An Enhanced Non-commutative Representation for Knowledge Graph Embedding with Adaptive Semantic Hierarchy

Capturing the composition patterns of relations is a vital task in knowledge graph completion. It also serves as a fundamental step towards multi-hop reasoning over learned knowledge. Previously, several rotation-based translational methods have been developed to model composite relations using the product of a series of complex-valued diagonal matrices. However, these methods tend to make several oversimplified assumptions on the composite relations, e.g., forcing them to be commutative, independent from entities and lacking semantic hierarchy. To systematically tackle these problems, we have developed a novel knowledge graph embedding method, named DensE, to provide an improved modeling scheme for the complex composition patterns of relations. In particular, our method decomposes each relation into an SO(3) group-based rotation operator and a scaling operator in the three dimensional (3-D) Euclidean space. This design principle leads to several advantages of our method: (1) For composite relations, the corresponding diagonal relation matrices can be non-commutative, reflecting a predominant scenario in real world applications; (2) Our model preserves the natural interaction between relational operations and entity embeddings; (3) The scaling operation provides the modeling power for the intrinsic semantic hierarchical structure of entities; (4) The enhanced expressiveness of DensE is achieved with high computational efficiency in terms of both parameter size and training time; and (5) Modeling entities in Euclidean space instead of quaternion space keeps the direct geometrical interpretations of relational patterns. Experimental results on multiple benchmark knowledge graphs show that DensE outperforms the current state-of-the-art models for missing link prediction, especially on composite relations.