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.

翻译：双曲空间已被证明能够生成层级结构的优越低维嵌入，这是欧几里得空间无法实现的。在此基础上，Ganea等人提出的蕴涵锥公式利用测地凸锥在双曲空间中嵌入偏序关系。然而，这些蕴涵锥由于通过黎曼空间中的切向量和指数映射等复杂概念定义，缺乏直观解释。本文提出影子锥这一创新框架，通过隐喻性光源与物体阴影（受日地月关系启发），为一般流形上定义偏序关系提供了物理直观的解释。影子锥包含两类主要锥体：本影锥与半影锥。实验结果表明，影子锥在WordNet、ConceptNet等多个数据集上展现出优异的表征与泛化能力，性能超越当前最优的蕴涵锥。我们的发现表明，影子锥提供了一种创新的通用方法，能够以几何方式编码偏序关系，从而更优地表征与分析具有层级结构的数据集。