Shadow Cones: Unveiling Partial Orders in Hyperbolic Space

Hyperbolic space has been shown to produce superior low-dimensional embeddings of hierarchical structures that are unattainable in Euclidean space. Building upon this, the entailment cone formulation of Ganea et al. uses geodesically convex cones to embed partial orderings in hyperbolic space. However, these entailment cones lack intuitive interpretations due to their definitions via complex concepts such as tangent vectors and the exponential map in Riemannian space. In this paper, we present shadow cones, an innovative framework that provides a physically intuitive interpretation for defining partial orders on general manifolds. This is achieved through the use of metaphoric light sources and object shadows, inspired by the sun-earth-moon relationship. Shadow cones consist of two primary classes: umbral and penumbral cones. Our results indicate that shadow cones offer robust representation and generalization capabilities across a variety of datasets, such as WordNet and ConceptNet, thereby outperforming the top-performing entailment cones. Our findings indicate that shadow cones offer an innovative, general approach to geometrically encode partial orders, enabling better representation and analysis of datasets with hierarchical structures.