将比例类比推广到非欧空间，统一了球面与概率分布等场景的类比推理
A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds
- 基于黎曼流形提出广义平行四边形法则，定义比例类比关系
- 在球面、形状空间和概率分布流形上验证类比有效性
- 为非欧空间中的类比推理提供统一数学框架，适合几何学习研究者
类比是形如‘a之于b犹如c之于d’的四元关系，通常记作 a : b :: c : d。比例类比通过特定约束定义有效类比。尽管比例类比在符号域和向量空间中已有研究，但在非欧几里得空间中应用受限。本文在黎曼流形上引入比例类比关系，扩展了欧氏空间中用于算术类比的平行四边形法则。我们在球面、形状空间及概率分布流形等多种流形上展示了该类比的应用效果。
原文摘要 · Abstract (English)
Analogies are quaternary relations of the form "a is to b as c is to d", usually denoted a : b :: c : d. This notion is formalized in particular with the notion of proportional analogy, which imposes some constraints on the valid analogies. Whereas proportional analogies have been studied mostly in symbolic domains and in vector spaces, their use is limited in non-Euclidean spaces. In this paper, we introduce a proportional analogy relation in Riemannian domains, extending the parallelogram rule used for arithmetic analogies in Euclidean spaces. We illustrate the introduced analogy on various manifolds, such as the sphere, shape spaces and manifolds of probability distributions.
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