A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

Analogical proportions link four items a, b, c, d by a relation stating that ``a is to b as c is to d", a, b, c, d being the formal representation of real world entities, ranging from simple numerical values to more complex structures such as profiles. Accordingly, $a, b, c, d$ could be atomic values like Boolean, nominal or numerical values, more generally vectors of such values, or even families of items represented by logical formulas. In this paper, we consider another representation setting, which is the probabilistic one. Precisely, the article proposes a study of {analogical} proportions between probabilities, whether they are simply between probability values, or between distributions (which requires the preservation of their normalization). More particularly, we study the properties of definitions based on arithmetic proportion, or on a combination of the former with geometric proportion, while other options are also discussed. Previous works have shown that when four profiles a, b, c, d, represented as vectors, form analogical proportions componentwise, it is likely that their classes form an analogical proportion also. This is the basis of an analogical proportion-based classification method that can produce accurate predictions. Similarly, in this paper, each profile is associated with a distribution describing the frequencies of the possible values of a discrete attribute of interest. We then discuss and experimentally investigate if the distributions associated to four profiles forming an analogical proportion themselves also form an analogical proportion.

Analogies are quaternary relations of the form "A is to B as C is to D". Among the various formalizations of analogical reasoning, proportional analogies provide an important axiomatic framework by characterizing valid analogies through a set of postulates. While proportional analogies have been extensively studied over Boolean, symbolic, and real-valued domains, their extension to probability distributions remains largely unexplored. In this paper, we introduce a notion of proportional analogy for probability distributions based on Bayesian updating. Our approach builds upon the idea that two distributions are related whenever one can be transformed into the other through Bayesian updating induced by a suitable set of observations. We investigate this framework for several standard members of the exponential family and discuss how it naturally extends to arbitrary probability distributions through Gaussian mixture approximations.

Four-term word analogies (A:B::C:D) are classically modeled geometrically as parallelograms: adding the vector B-A+C produces D. Recent work suggests that this model poorly captures how humans produce analogies, with simple local-similarity heuristics often providing a better account (Peterson et al., 2020). But does the parallelogram model fail because it is a bad model of analogical relations, or because people are not very good at generating relation-preserving analogies? We compared human and large language model (LLM) analogy completions on the set of problems from Peterson et al. (2020). We find that LLM-generated analogies are reliably judged as better than human-generated ones, and are also more consistent with parallelograms in a distributional embedding space. Crucially, we show that the improvement over human analogies is driven by greater parallelogram alignment and reduced reliance on accessible words rather than enhanced sensitivity to local similarity. Finally, fine-tuning GloVe to better satisfy the parallelogram constraint makes the model's top-ranked candidates more likely to be the completions humans and LLMs actually produced, and improves its prediction of human ratings. Overall, these results provide support for the parallelogram model of word analogies.