A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

Excerpt

arXiv:2608.14220v1 Announce Type: new 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 Riem