Shadow Cones: A Generalized Framework for Partial Order Embeddings
arxiv(2023)
摘要
Hyperbolic space has proven to be well-suited for capturing hierarchical
relations in data, such as trees and directed acyclic graphs. Prior work
introduced the concept of entailment cones, which uses partial orders defined
by nested cones in the Poincaré ball to model hierarchies. Here, we introduce
the “shadow cones" framework, a physics-inspired entailment cone construction.
Specifically, we model partial orders as subset relations between shadows
formed by a light source and opaque objects in hyperbolic space. The shadow
cones framework generalizes entailment cones to a broad class of formulations
and hyperbolic space models beyond the Poincaré ball. This results in clear
advantages over existing constructions: for example, shadow cones possess
better optimization properties over constructions limited to the Poincaré
ball. Our experiments on datasets of various sizes and hierarchical structures
show that shadow cones consistently and significantly outperform existing
entailment cone constructions. These results indicate that shadow cones are an
effective way to model partial orders in hyperbolic space, offering physically
intuitive and novel insights about the nature of such structures.
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