New Proportional Analogy Framework for Riemannian Manifolds

A new research paper introduces a generalized parallelogram rule for proportional analogies on Riemannian manifolds, extending the concept beyond Euclidean spaces. The paper, titled 'A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds,' is authored by researchers in the field of artificial intelligence and was submitted to arXiv. It formalizes proportional analogies, which are quaternary relations of the form 'a is to b as c is to d,' and imposes constraints on valid analogies. While proportional analogies have been studied in symbolic domains and vector spaces, their use in non-Euclidean spaces has been limited. This work addresses that gap by defining a proportional analogy relation in Riemannian domains, using a generalized parallelogram rule. The authors illustrate the approach on various manifolds, including the sphere, shape spaces, and manifolds of probability distributions. The paper is categorized under Computer Science > Artificial Intelligence and was submitted on arXiv with the identifier 2608.14220. The research contributes to the theoretical foundations of analogical reasoning in geometric settings, with potential applications in machine learning and data analysis on non-Euclidean data.