A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

2026-08-14

AI Summary

This study addresses the limitation of proportional analogy definitions in non-Euclidean spaces by generalizing the Euclidean parallelogram law to Riemannian manifolds. We propose a generalized proportional analogy framework applicable to spherical, shape, and probability distribution manifolds, thereby overcoming theoretical bottlenecks in analogical reasoning under non-Euclidean geometry and achieving a critical extension from Euclidean space to Riemannian domains. Experimental results across multiple manifolds validate the effectiveness of the proposed approach. Consequently, this work establishes a robust theoretical foundation and provides a universal solution for structured analogy modeling in non-Euclidean spaces, significantly advancing geometric reasoning capabilities beyond flat geometries.

Abstract

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.

Problem

• Proportional Analogy
• Riemannian Manifolds
• Non-Euclidean Spaces

Innovation

• Proportional Analogy
• Riemannian Manifolds
• Parallelogram Rule
• Non-Euclidean Geometry