黎曼流形上比例类比的广义平行四边形法则
A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds
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中文总结 AI 辅助
本文针对非欧空间中比例类比应用受限的问题，提出黎曼域的比例类比关系，扩展欧氏空间平行四边形法则，并在多种流形上验证该类比的有效性。
中文摘要 AI 辅助
类比是形如“a之于b如同c之于d”的四元关系，通常记为a: b:: c: d，该概念经比例类比形式化，对有效类比施加约束。比例类比多在符号域和向量空间中研究，其在非欧空间中的应用受限。本文提出黎曼域中的比例类比关系，扩展欧氏空间算术类比所用的平行四边形法则，并在球面、形状空间、概率分布流形等多种流形上对所提类比进行了示例说明。
英文摘要
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.
