Analogical reasoning is a unique characteristic of human cognition, that consists in transferring properties from one problem to the other based on their similarities and dissimilarities. It corresponds typically to statements of the form “Berlin is to Germany as Paris is to France”. Even though this form of reasoning is elementary in our cognition, it is particularly challenging to replicate on machines: indeed, it involves identification of structures, estimation of relevance and creation of concepts, which are still difficult tasks, even for modern AIs.

An important mathematical characterization of analogies, called “proportional analogy”, defines analogies as quaternary relations constrained by a set of regularity constraints. Proportional analogies have been mostly studied in Boolean or Euclidean domains, but more rarely in Riemannian manifolds. Defining analogies in manifolds has important practical applications, in particular in machine learning through information geometry, or in computer graphics.