A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

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.