文章背景与核心概要

比例类比（Proportional analogies）通常表现为“\(a\)之于\(b\)犹如\(c\)之于\(d\)”（记作 \(a : b :: c : d\)）的四元关系，在符号推理和欧几里得向量空间中得到了广泛的研究。然而，将这种直观的类比推理扩展到更复杂的非欧几里得几何（如弯曲流形）领域，长期以来一直面临着形式化定义和计算上的挑战。

为了弥补这一研究空白，本文作者 Pierre-Alexandre Murena 和 Marcelo Hartmann 提出了一种在黎曼流形（Riemannian domains）上定义比例类比的新颖框架。该框架成功将欧几里得空间中用于算术类比的传统“平行四边形法则”推广到了黎曼几何中。通过在球面、形状空间以及概率分布流形等多种复杂流形上的具体应用与展示，该研究所提出的广义方法展现出了强大的理论普适性和实用价值。

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)

摘要 (Summary)

本文引入了一个新颖的框架，用于在非欧几里得几何中定义比例类比——传统上表示为形式为“\(a\)之于\(b\)犹如\(c\)之于\(d\)”的四元关系（\(a : b :: c : d\)）。虽然比例类比已在符号空间和向量空间中得到了广泛探讨，但其在非欧几里得域中的应用仍然有限。为了填补这一空白，作者在黎曼域中提出了一种比例类比关系，成功推广了欧几里得空间中用于算术类比的传统平行四边形法则。这种广义方法的有效性在各种流形（包括球面、形状空间和概率分布流形）上得到了证明。

This paper introduces a novel framework for defining proportional analogies—traditionally expressed as quaternary relations of the form "\(a\) is to \(b\) as \(c\) is to \(d\)" (\(a : b :: c : d\))—within non-Euclidean geometries. While proportional analogies have been extensively explored in symbolic and vector spaces, their application to non-Euclidean domains remains limited. To bridge this gap, the authors propose a proportional analogy relation in Riemannian domains that successfully extends the traditional parallelogram rule used for arithmetic analogies in Euclidean spaces. The effectiveness of this generalized approach is demonstrated across various manifolds, including spheres, shape spaces, and manifolds of probability distributions.

元数据 (Metadata)

Field	Details
Authors	Pierre-Alexandre Murena, Marcelo Hartmann
Primary Subject	Artificial Intelligence (cs.AI)
arXiv Identifier	arXiv:2608.14220 [cs.AI]
Publication Date	14 August 2026
DOI	10.48550/arXiv.2608.14220
License	Creative Commons Attribution 4.0 International

摘要原文 (Abstract)

类比是形式为“a之于b犹如c之于d”的四元关系，通常记作 \(a : b :: c : d\)。这种概念特别通过比例类比的概念进行形式化，该概念对有效的类比施加了某些约束。鉴于比例类比主要在符号域和向量空间中进行研究，其在非欧几里得空间中的应用受到限制。在本文中，我们在黎曼域中引入了一种比例类比关系，推广了欧几里得空间中用于算术类比的平行四边形法则。我们在各种流形（如球面、形状空间和概率分布流形）上说明了引入的类比。

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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