A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

Pierre-Alexandre Murena; Marcelo Hartmann

cs.AIAug 14, 2026

A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

Authors: Pierre-Alexandre Murena, Marcelo Hartmann

Organizations: Hamburg University of Technology (TUHH), Research Group on Human-Centric Machine Learning, Germany · Department of Computer Science, University of Helsinki, Finland

Abstract

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.

Figures & tables

Figure 1: Analogical equations solved by our generalized parallelogram rule on Riemannian manifolds. Domain 1 corresponds to Symmetric Positive Definite Matrices, with results presented in Corollary 11 . Domain 2 is Kendall’s space of triangular shapes, with results presented in Corollary 8 . Domain 3 is the space of 3D triangular meshes, using approximated geodesics and logarithmic maps. Domain 4 is the simplex of probability distributions, endowed with the Aitchison metric, with results presented in Proposition 15 . The source code to generate these examples is in supplementary material, and will be published upon acceptance.

Panel of Figure 2 — Figure 2: Analogy $a:b::c:d$ holds since the geodesics $\gamma_{a,d}$ from $a$ to $d$ and $\gamma_{b,c}$ from $b$ to $c$ intersect at their midpoint.

Figure 3: An example of a rating profile transfer between two cohorts (

25-34

men and

<25

women) and two categories (action and film noir). A rating profile is represented as a distribution

p\in\Delta^{4}

Experiment	LOO Genre	LOO Add.	Fisher-Rao	Aitchison	Prade
Age	$1.323_{[1.108,\,1.560]}$	$0.366_{[0.259,\,0.487]}$	$\mathbf{0.274}_{[0.190,\,0.373]}$	$0.314_{[0.210,\,0.451]}$	$0.294_{[0.208,\,0.393]}$
Age + Gender	$1.731_{[1.449,\,2.027]}$	$0.842_{[0.641,\,1.064]}$	$\mathbf{0.679}_{[0.523,\,0.863]}$	$0.708_{[0.546,\,0.881]}$	$0.686_{[0.531,\,0.863]}$
Occupation	$1.962_{[1.690,\,2.244]}$	$1.134_{[0.919,\,1.408]}$	$0.956_{[0.786,\,1.164]}$	$\mathbf{0.931}_{[0.766,\,1.128]}$	$0.990_{[0.822,\,1.181]}$
Single Genre	$4.360_{[3.538,\,5.216]}$	$3.432_{[2.498,\,4.527]}$	$\mathbf{2.649}_{[1.984,\,3.411]}$	$2.865_{[2.152,\,3.655]}$	$2.677_{[1.961,\,3.462]}$
Independent Users	$2.431_{[2.079,\,2.823]}$	$2.040_{[1.667,\,2.464]}$	$\mathbf{1.899}_{[1.554,\,2.291]}$	$1.949_{[1.586,\,2.355]}$	$1.904_{[1.568,\,2.272]}$
Occupation + Genre + Decade	$7.378_{[6.993,\,7.779]}$	$8.138_{[7.701,\,8.585]}$	$\mathbf{7.350}_{[6.984,\,7.738]}$	$7.383_{[7.003,\,7.767]}$	$7.368_{[6.981,\,7.745]}$

Table 1: Mean Jensen-Shannon divergence (

\times 10^{-3}

) on the MovieLens preference transfer task. Lower is better. Values in brackets denote 95% bootstrap confidence intervals.

Analogical Reasoning

Analogical reasoning studies how systems transfer relations and solutions between situations with shared structure, enabling inference beyond direct examples.

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From numerical proportions to analogical proportions between probabilities

Henri Prade, Gilles Richard

IRIT · Cr Rose Dieng-Kuntz · Toulouse, 31400, France

Analogical Reasoning

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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.

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