Physics

Pure mathematics

Mathematics

Lorentz transformation

Hyperbolic space

Operating system

Image (mathematics)

Similarity (geometry)

Algebra over a field

Artificial intelligence

Hyperbolic geometry

Differential geometry

Space (punctuation)

Computer science

Theoretical computer science

Poincaré conjecture

Embedding

Classical mechanics

We are concerned with the discovery of hierarchical relationships from large-scale unstructured similarity scores. For this purpose, we study different models of hyperbolic space and find that learning embeddings in the Lorentz model is substantially more efficient than in the Poincare-ball model. We show that the proposed approach allows us to learn high-quality embeddings of large taxonomies which yield improvements over Poincare embeddings, especially in low dimensions. Lastly, we apply our model to discover hierarchies in two real-world datasets: we show that an embedding in hyperbolic space can reveal important aspects of a company's organizational structure as well as reveal historical relationships between language families.

Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry