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.

本文提出了一种基于影子锥的框架来定义广义流形上的部分排序，该框架通过使用隐喻性的光源和物体阴影来实现直观的物理解释。这种方法在 WordNet 和 ConceptNet 等不同数据集上表现出强大的表征和泛化能力，并且比其他方法表现更优秀，为分析具有分层结构的数据集提供了更好的编码方法。