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A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

This paper introduces a generalized proportional analogy relation for Riemannian manifolds by extending the Euclidean parallelogram rule to non-Euclidean spaces, demonstrating its applicability on diverse domains such as spheres, shape spaces, and probability distribution manifolds.

Original authors: Pierre-Alexandre Murena, Marcelo Hartmann

Published 2026-08-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Pierre-Alexandre Murena, Marcelo Hartmann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a riddle that looks like this: "Apple is to Fruit as Carrot is to...?" Your brain instantly knows the answer is "Vegetable." You didn't need a calculator; you just understood the relationship between the first two items and applied it to the third. In the world of computers, this is called an "analogy," and it's a superpower for artificial intelligence. For a long time, scientists taught computers to solve these riddles using flat, grid-like maps (called Euclidean spaces), where you can just draw a straight line or a perfect parallelogram to find the missing piece. It works great for simple lists of words or numbers.

But the real world isn't flat. Think of the Earth: it's a sphere. If you try to draw a straight line on a globe, it curves. Or think of a shape-shifting robot or a cloud of probabilities; these exist on complex, curved surfaces called "manifolds." When computers try to solve analogies on these curved surfaces using old, flat rules, things break. The straight lines don't connect, and the answers get messy. This paper asks a big question: How can we teach computers to solve "A is to B as C is to D" riddles when the world they live in is curved, twisted, and full of bumps?

The authors, Pierre-Alexandre Murena and Marcelo Hartmann, have come up with a clever new way to do this. Instead of trying to force a curved world into a flat box, they decided to use the curves themselves to find the answer. They realized that in a flat world, a parallelogram is defined by its sides being parallel. But on a curved surface, "parallel" is tricky. So, they changed the definition. They looked at the center of the shape. In a flat square, the diagonals cross right in the middle. The authors proved that if you find the "geodesic midpoint" (the exact center point along the shortest curved path) between two items, you can use that to solve the analogy, even on a sphere, a shape-shifting mesh, or a cloud of data.

Here is how their new rule works in plain English. Imagine you have four points on a curved surface: A, B, C, and D. In the old flat world, you'd say, "The distance from A to B is the same as C to D." On a curved world, that's hard to measure. The authors say, "Forget the sides. Look at the middle." If you walk halfway from A to D, and you walk halfway from B to C, you should land on the exact same spot. If those two "midpoint" meetings happen at the same place, then A, B, C, and D form a perfect analogy. It's like saying, "The halfway point between the start and finish of the first journey is the same as the halfway point of the second journey."

This might sound like a small tweak, but it's a huge deal because it works everywhere. The authors tested their idea on all sorts of weird, curved places. They tried it on a sphere (like the Earth), on 3D shapes (like a dog turning into a cow), and even on clouds of probability (like predicting movie ratings). In every case, their "midpoint rule" found the missing piece of the puzzle correctly. They showed that this method is "robust," meaning if you nudge the starting points just a tiny bit, the answer doesn't crash and burn; it stays close to the right answer. This is crucial for real-world AI, where data is often noisy or imperfect.

One of the coolest parts of their work is how it handles different types of "curved" data. For example, when dealing with shapes (like 3D models of animals), they showed that you can take a deformation (a stretch or twist) from one animal and apply it to another using this rule. If you have a dog in a sitting pose and a dog in a standing pose, you can figure out how to turn a cow from sitting to standing, even though cows and dogs are different. They also tested this on movie ratings. If you know how "teenagers" rate action movies versus horror movies, and you know how "adults" rate action movies, you can use their new rule to guess how "adults" will rate horror movies. In their tests, this geometric approach was often better than older methods at predicting these preferences.

The paper also makes a clear distinction about where this works and where it might get tricky. They proved that on certain smooth, symmetric surfaces (like spheres or hyperbolic spaces), the answer is unique and easy to calculate. However, on some surfaces, like the sphere, if two points are exactly opposite each other (antipodal), there might be more than one "middle" point, which means there could be more than one valid answer to the riddle. They didn't just guess this; they provided the mathematical formulas to calculate the exact answer for spheres, hyperbolic spaces, and even complex matrices used in machine learning.

What they didn't do is claim that this solves every problem in AI. They explicitly noted that their method relies on the data existing on a specific type of curved surface called a "Riemannian manifold." If the data lives on a weird, non-smooth surface that doesn't fit these rules, their method might not apply directly. They also pointed out that while their method works beautifully for shapes and probabilities, it's a new tool that needs to be tested more in real-world applications like transfer learning (teaching a computer one task to help it with another) or meta-learning.

In the end, this paper is about giving AI a better map. For years, computers tried to navigate the complex, curved world of data using a flat, 2D map, which led to getting lost. Murena and Hartmann have drawn a new map that respects the curves. By focusing on the "middle" of the journey rather than the straight lines, they've shown that computers can finally solve analogies in the real, twisted, beautiful world we actually live in. It's a step toward AI that doesn't just memorize facts, but truly understands the relationships between things, whether those things are words, shapes, or movie tastes.

Problem Statement

Proportional analogies, formalized as quaternary relations $a:b::c:d$ (" $a$ is to $b$ as $c$ is to $d$ "), are fundamental to human cognition and machine learning validation. While the concept is well-established in symbolic domains and Euclidean vector spaces—where the arithmetic parallelogram rule ( $b-a = d-c$ ) serves as a valid proportional analogy—its application is limited in non-Euclidean spaces. Previous attempts to extend the parallelogram rule to Riemannian manifolds (e.g., via geodesic shooting and parallel transport) have faced two primary limitations: the direct generalization fails to satisfy the axioms of proportional analogies (specifically, the solution to $a:b::c:x$ differs from $a:c::b:x$ ), and general constructions do not guarantee continuity properties.

Methodology

The paper proposes a new, intrinsic definition of proportional analogies on Riemannian manifolds based on geodesic midpoints rather than parallel displacements. This approach draws on the geometric insight that in Euclidean space, a parallelogram is defined by the coincidence of the midpoints of its diagonals.

Core Construction:
For a Riemannian manifold $(\mathcal{M}, g)$ , the relation $a:b::c:d$ holds if and only if the geodesic midpoint between $a$ and $d$ coincides with the geodesic midpoint between $b$ and $c$ .

Geodesically Convex Case: If the manifold is geodesically convex (unique geodesics exist between any two points), the relation is defined by:
$\exp_a\left(\frac{1}{2}\log_a(d)\right) = \exp_b\left(\frac{1}{2}\log_b(c)\right)$
This formulation ensures the relation is a strong proportional analogy.
General Case: For non-convex manifolds where geodesics are not unique, the definition is generalized to:
$\exists v \in T_a\mathcal{M}, \exists w \in T_b\mathcal{M} \text{ such that } c=\exp_b(w), d=\exp_a(v), \text{ and } \exp_a(v/2) = \exp_b(w/2)$

Theoretical Framework:
The authors analyze the properties of this construction, including:

Axiomatic Satisfaction: Proving the relation satisfies symmetry, exchange of means, and reflexivity.
Isometry Invariance: Demonstrating that the analogy is preserved under Riemannian isometries.
Robustness: Establishing that the analogy is robust (continuous under small perturbations) on Hadamard manifolds and Riemannian symmetric spaces.
Realizability in $\mathbb{R}^n$ : Showing that for $n \geq 2$ , any four distinct points can form an analogy under a suitable metric, whereas for $n=1$ , the intervals formed by the points must be nested.

Key Contributions

Intrinsic Definition: The introduction of a proportional analogy relation on Riemannian manifolds based on geodesic midpoints, which satisfies the axioms of proportional analogies and is entirely intrinsic (relying only on exponential and logarithmic maps).
Theoretical Properties: The establishment of key properties, including invariance under isometries, robustness, and explicit characterizations on geodesically convex and symmetric spaces.
Extension of Lepage and Couceiro (2024): Proving that any quadruple of pairwise distinct points in $\mathbb{R}^n$ ( $n \geq 2$ ) can be made analogous under a suitable metric, and providing ordering conditions for $n=1$ .
Versatility Across Domains: The application of the framework to diverse geometric domains, providing closed-form solutions or algorithms for:
- Symmetric Spaces: Spheres ( $\mathbb{S}^n$ ), Hyperbolic spaces ( $\mathbb{H}^n$ ), and Symmetric Positive Definite (SPD) matrices.
- Shape Spaces: Kendall's space of triangular shapes and 3D triangular meshes.
- Probability Manifolds: The simplex of categorical distributions under both the Fisher-Rao and Aitchison metrics.

Results and Applications

The paper illustrates the framework through several specific applications and theoretical derivations:

SPD Matrices: For the manifold of symmetric positive definite matrices, the analogical equation $A:B::C:X$ admits a unique solution involving the geometric mean of matrices: $X = (B\#C)A^{-1}(B\#C)$ .
Shape Spaces: The method is applied to 3D triangular meshes (using the Morphomatics library) to transfer deformations. In an experiment transferring a pose change from a dog mesh to a cow mesh, the framework successfully generated the target mesh in approximately 2 seconds, demonstrating generalization to complex geometric domains.
Categorical Distributions (Preference Transfer): The authors evaluated the framework on the MovieLens 1M dataset for a preference transfer task. The goal was to predict the rating distribution of a target demographic group for a target movie category, given three reference distributions.
- Metrics: Performance was measured using Mean Jensen-Shannon divergence.
- Findings: The analogical approaches (Fisher-Rao and Aitchison metrics) consistently outperformed non-analogical baselines (leave-one-out predictors and additive models) across six experimental conditions of increasing difficulty (varying by age, gender, occupation, and genre).
- Comparison: The Fisher-Rao metric achieved the lowest divergence in five of six settings, while the Aitchison metric performed best for occupation-based cohorts. Both outperformed the arithmetico-geometric analogy baseline.

Significance

The paper claims to provide a unified geometric framework for extending proportional analogies beyond symbolic domains and Euclidean vector spaces. By characterizing analogies through geodesic midpoints, the authors demonstrate that a single intrinsic construction naturally extends analogical reasoning to complex structures like manifolds of probability distributions, shape spaces, and SPD matrices. The work establishes that these analogies are not only theoretically sound (satisfying axioms and robustness) but also practically effective, as shown by the successful transfer of preference distributions and geometric deformations. The authors suggest this framework could serve as a foundation for future developments in transfer learning, meta-learning, and geometric data augmentation.

Problem Statement

Methodology

Key Contributions

Results and Applications

Significance

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