Combinatorics

Discrete mathematics

Graph

Topology (electrical circuits)

Pure mathematics

Voltage graph

Mathematics

Manifold (fluid mechanics)

Hyperbolic space

Gene

Euclidean geometry

Engineering

Artificial intelligence

Topological graph theory

Hyperbolic geometry

Chemistry

Curvature

Line graph

Transformation (genetics)

Geometry

Transformation geometry

Computer science

Algebraic geometry

Theoretical computer science

Embedding

Biochemistry

Mechanical engineering

Hyperbolic tree

Recent knowledge graph (KG) embeddings have been advanced by hyperbolic geometry due to its superior capability for representing hierarchies. The topological structures of real-world KGs, however, are rather heterogeneous, i.e., a KG is composed of multiple distinct hierarchies and non-hierarchical graph structures. Therefore, a homogeneous (either Euclidean or hyperbolic) geometry is not sufficient for fairly representing such heterogeneous structures. To capture the topological heterogeneity of KGs, we present an ultrahyperbolic KG embedding (UltraE) in an ultrahyperbolic (or pseudo-Riemannian) manifold that seamlessly interleaves hyperbolic and spherical manifolds. In particular, we model each relation as a pseudo-orthogonal transformation that preserves the pseudo-Riemannian bilinear form. The pseudo-orthogonal transformation is decomposed into various operators (i.e., circular rotations, reflections and hyperbolic rotations), allowing for simultaneously modeling heterogeneous structures as well as complex relational patterns. Experimental results on three standard KGs show that UltraE outperforms previous Euclidean, hyperbolic, and mixed-curvature KG embedding approaches.

Ultrahyperbolic Knowledge Graph Embeddings