# A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2608.14220 · HTML: https://mlsanity.com/paper/2608.14220/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 24 references that are in the map.
- First posted on arXiv 14 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Riemannian Dictionary Learning and Sparse Coding for Positive Definite Matrices](https://mlsanity.com/paper/1507.02772/) (2015)
- [Geometric Mean Metric Learning](https://mlsanity.com/paper/1607.05002/) (2016)
- [Any four real numbers are on all fours with analogy](https://mlsanity.com/paper/2407.18770/) (2024)

## Abstract (arXiv)

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.