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A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds

Pierre-Alexandre Murena, Marcelo Hartmann

arXiv:2608.14220v1cs.AI

TL;DR

Proportional analogies are difficult to use beyond symbolic and Euclidean settings because curvature disrupts the classical parallelogram construction. This paper replaces Euclidean diagonal midpoints with intrinsic geodesic midpoints, yielding a Riemannian construction that satisfies proportional-analogy axioms and applies across diverse geometric domains.

  • Problem

    Curvature makes transporting tangent vectors an inadequate proportional-analogy construction on Riemannian manifolds because reordered analogies can produce different solutions.

  • Method

    The paper defines proportional analogies intrinsically through shared geodesic midpoints, using exponential and logarithmic maps on Riemannian manifolds.

  • Results

    The proposed relation satisfies proportional-analogy axioms, has theoretical properties including isometry invariance and robustness, and is illustrated across diverse geometric domains.

  • Takeaways & Limitations

    Geodesic-midpoint characterization provides a single intrinsic construction for analogical reasoning on arbitrary Riemannian manifolds.

  • Takeaways & Limitations

    The equivalence of the two proportional-analogy definitions requires global geodesic convexity, or uniqueness of geodesics at the points considered.

Abstract

from arXiv · show

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.

Introduction

The section motivates proportional analogies as a broadly applicable model and identifies the lack of valid, continuous extensions of the Euclidean parallelogram rule to manifold-valued domains. It introduces a geodesic-midpoint formulation on Riemannian manifolds, with axiomatic and theoretical guarantees.

  • Motivation: Analogies are quaternary relations whose solution requires finding d, a capability linked to human cognition and used to validate machine-learning methods.Word embeddings such as word2vec are cited as an early example of analogies validating machine-learning methods.
  • Background: Proportional analogies axiomatize when a:b::c:d holds and apply across many domains, including symbolic settings and vector spaces.The Euclidean parallelogram rule is identified as a valid proportional analogy.
  • Problem: Existing manifold extensions face limitations: geodesic shooting and parallel transport do not yield valid proportional analogies, while general constructions lack continuity guarantees.These limitations motivate a different geometric characterization.
  • Contribution: The proposed framework replaces parallel displacements with intrinsic geodesic midpoints, extending the Euclidean characterization that parallelogram diagonals share a midpoint.The construction is presented as applying to arbitrary Riemannian manifolds.
  • Contribution: The paper defines proportional analogies intrinsically on Riemannian manifolds, proves the proportional-analogy axioms, and establishes invariance, robustness, and characterizations on geodesically convex and symmetric spaces.The stated theoretical results include extensions of prior work on pairwise-distinct quadruples.

Proportional Analogies

Proportional analogies are quaternary relations defined by symmetry, exchange of the means, and reflexivity. In vector spaces, the parallelogram rule provides a proportional, or arithmetic, analogy, while a center-based generalized-mean formulation offers an alternative.

  • Proportional Analogies: A proportional analogy a:b::c:d satisfies symmetry, exchange of the means, and reflexivity for all a, b, c, and d.These properties generate eight equivalent forms of the relation.
  • Proportional Analogies: A proportional analogy is strong when a:b::a:x implies x=b for every a and b.Strongness ensures uniqueness in this repeated-first-pair analogical equation.
  • Proportional Analogies: In a vector space, b−a=d−c defines a proportional analogy called arithmetic proportion, corresponding to the parallelogram rule.The rule compares corresponding parallel sides, equivalently expressing that b differs from a as d differs from c.
  • Proportional Analogies: A center-based formulation declares a:b::c:d when the generalized means m_p(a,c) and m_p(b,d) coincide, yielding a proportional analogy.This perspective characterizes the parallelogram through coincident segment middles rather than parallel sides.

Riemannian Geometry

The section introduces the geometric structures underlying Riemannian manifolds, from differentiable charts and tangent spaces to geodesics, exponential and logarithmic maps, and arclength. These constructions provide the framework for representing points and directions through geodesic motion.

  • A topological manifold is locally homeomorphic to an open subset of ℝ^d, and differentiability requires smooth transitions between charts.
  • Tangent vectors are equivalence classes of curves with first-order contact, forming the d-dimensional tangent space T_pℳ at each point.
  • A smoothly varying positive inner product on tangent spaces defines a Riemannian manifold (ℳ, g).
  • Geodesics satisfy ∇_{γ̇}γ̇ = 0, and each initial point p and tangent vector V determine a unique geodesic.
  • The exponential map sends an initial velocity V to the endpoint of its geodesic, while geodesic convexity enables the inverse logarithmic map log_p(q).

Generalized Parallelogram

The paper generalizes the parallelogram rule to Riemannian manifolds using geodesics, midpoint intersection, and exponential maps. It characterizes when the analogy is strong, invariant, solvable, robust, or realizable under geometric conditions.

  • Generalized Parallelogram: The construction replaces Euclidean differences with logarithmic-map tangent vectors transported between manifold points before shooting a geodesic.This direct transport procedure may fail in general, motivating an alternative characterization.
  • Generalized Parallelogram: A strong proportional analogy is characterized by geodesics from a to d and from b to c intersecting at their common midpoint.Proposition 6 provides the alternative characterization on geodesically convex Riemannian manifolds with a connection.
  • Generalized Parallelogram: When geodesics are nonunique, the generalized relation need not be strong; the original and generalized analogies coincide globally only under geodesic convexity.They coincide more locally when the relevant geodesic uniqueness condition holds.
  • Generalized Parallelogram: Riemannian isometries preserve the proportional analogy because they preserve geodesics, exponential maps, and their midpoint intersections.Thus an analogy on one Riemannian manifold transfers to its isometric image on another.
  • Generalized Parallelogram: On Hadamard manifolds of positive dimension, the Proposition 6 analogy is robust; in Euclidean spaces, realizability differs between one and higher dimensions.On R, realizability requires nested intervals, whereas for R^n with n≥2 some metric always realizes the analogy, reflecting diffeomorphism-group transitivity.

Proportional Analogies on Riemannian · Symmetric Spaces

The paper characterizes proportional analogies on Riemannian symmetric spaces through geodesic symmetries about midpoints. It derives consequences for Euclidean, spherical, hyperbolic, and symmetric positive-definite matrix geometries, including uniqueness and robustness properties.

  • Symmetric Spaces: A connected Riemannian manifold is symmetric when every point p admits an isometry s_p fixing p with derivative D(s_p)_p = −Id.This symmetry reflects every geodesic through p by transforming γ(t) into γ(−t).
  • Symmetric Spaces: On a Riemannian symmetric space, a : b ::_g c : d holds exactly when some midpoint m of b and c satisfies d = s_m(a).The midpoint-centered geodesic symmetry provides the generalized parallelogram rule.
  • Symmetric Spaces: In Euclidean space, s_m(a) = 2m − a and m = (b + c)/2, recovering the parallelogram rule d = c + b − a.Thus the symmetric-space characterization extends the standard arithmetic analogy rule.
  • Symmetric Spaces: The proportional analogy is robust at every valid quadruple on any positive-dimensional Riemannian symmetric space.Every neighborhood of a valid quadruple contains another valid quadruple, and symmetry—not the Hadamard property—is sufficient.
  • Analogies on Spheres: On the sphere, when b and c are not antipodal, both midpoints induce the same geodesic symmetry, so the analogical equation has a unique solution.The sphere’s geodesic symmetry is s_m(x) = 2⟨m, x⟩m − x.
  • Analogies on Spheres: When b = −c on the sphere, the midpoint is nonunique and the solution set is {d ∈ S^n : ⟨d, b⟩ = −⟨a, b⟩}.For n = 2, solutions have azimuthal angle θ = π − θ_a.
  • Analogies on Hyperbolic Spaces: Hyperbolic space is a Hadamard manifold with unique geodesic midpoints, so its analogical equation admits a unique solution.Its hyperboloid model is a Riemannian symmetric space with s_m(x) = −2⟨x, m⟩_L m − x.
  • Analogies on Symmetric Positive Definite Matrices: Under affine-invariant metrics, SPD_n is a Riemannian symmetric space, and A : B :: C : X admits the unique solution stated in Corollary 8.The midpoint between B and C is B#C = B^1/2(B^−1/2CB^−1/2)^1/2B^1/2.

Proportional Analogies on Shape Spaces

The paper extends proportional analogies from Kendall’s planar-triangle shape space, which is isometric to the sphere, to three-dimensional triangle meshes. On meshes with corresponding topology, the analogy solver transfers deformations while preserving the target shape’s geometric characteristics.

  • Shape-space formulation: Shape spaces represent geometric objects independently of translation, rotation, and scaling, using quotient Riemannian spaces with intrinsic distances, geodesics, and means.Landmark configurations are identified under nuisance transformations, while continuous shapes can likewise be modeled through quotient spaces under diffeomorphism actions.
  • Planar triangles: Kendall’s planar-triangle shape space is obtained by centering, unit-size normalization, and quotienting preshapes by planar rotations, yielding a space isometric to the sphere 𝕊2.Therefore, the sphere analogy result established in Corollary 5 applies directly to planar triangles.
  • Three-dimensional meshes: The framework extends analogical reasoning to three-dimensional triangle meshes as a demonstration of generalization to geometric deformation transfer.The objective is to show that the proposed analogy relation naturally extends to this challenging geometric setting rather than compete with specialized deformation-transfer methods.
  • Three-dimensional meshes: Given meshes A, B, and C, the Morphomatics-based implementation solves the analogical equation for a missing mesh D on a Riemannian manifold of fundamental coordinates.The SMALR animal meshes share identical triangulation and one-to-one vertex correspondence; otherwise, preprocessing would be needed to establish correspondences.
  • Three-dimensional meshes: Approximately 2 s: in a dog-to-cow example, the computed mesh D transfers deformation A→B to the cow while preserving its geometric characteristics.The example illustrates successful transfer and low execution time, although it is not competitive with dedicated deformation-transfer methods.

Proportional Analogies on Categorical · Distributions

The section develops proportional analogies for categorical distributions using Fisher-Rao and Aitchison Riemannian geometries. On MovieLens preference transfer, these analogies outperform leave-one-out baselines across increasingly difficult settings, with Fisher-Rao generally strongest.

  • Distributions: The n-dimensional categorical-distribution space is represented by the simplex, where the section considers Fisher-Rao and Aitchison Riemannian geometries.Non-Riemannian geometries such as Hilbert geometry are outside the paper’s scope.
  • Fisher-Rao Metric: The Fisher-Rao simplex is mapped isometrically to the positive orthant of a sphere of radius 2, enabling the generalized parallelogram construction.The mapping is 𝜙(p) = (2√p_1, …, 2√p_n).
  • Fisher-Rao Metric: Under Fisher-Rao geometry, the analogical equation has a unique solution exactly when r_i > 0 for every component i.When solvable, the solution is obtained from the spherical construction and mapped back to the simplex.
  • Fisher-Rao Metric: Fisher-Rao analogies are not always solvable because the simplex is not geodesically complete and geodesics can leave the manifold in finite time.This prevents some geodesic symmetries from being defined globally.
  • Aitchison Metric: The Aitchison metric is the Euclidean pullback under the centered log-ratio map, making the simplex isometric to Euclidean space and guaranteeing a unique analogy solution.The clr image is the zero-sum hyperplane H.
  • Application: The MovieLens preference-transfer task predicts a target cohort’s rating distribution for a target category from three observed cohort-category distributions with symmetric Dirichlet smoothing.The evaluation compares Fisher-Rao, Aitchison, and arithmetico-geometric analogies against average-category and additive marginal baselines using Jensen-Shannon divergence.
  • Application: Across six increasingly difficult settings, analogical methods generally outperform both leave-one-out baselines, while Fisher-Rao is best in five experiments and Aitchison is best for occupation-based transfer.Analogical models retain their baseline advantage in the first five settings as Jensen-Shannon divergence increases with task difficulty.

Conclusion

The paper introduces an intrinsic proportional analogy relation on Riemannian manifolds using geodesic midpoints, establishes its theoretical properties, and derives closed forms on important manifolds. It provides a unified geometric framework while identifying extensions, efficient solvers, and empirical analogy-based learning as future directions.

  • Conclusion: The proposed relation uses geodesic midpoints to define proportional analogies intrinsically while reflecting the underlying manifold geometry.It satisfies the fundamental properties of proportional analogies.
  • Conclusion: Equivalent formulations hold on geodesically convex manifolds, and the relation is invariant under isometries.These are among the theoretical characterizations established for the analogy relation.
  • Conclusion: On Riemannian symmetric spaces, the analogy has a particularly simple form with closed-form characterizations for spheres, hyperbolic spaces, symmetric positive definite matrices, shape spaces, and probability distributions.The result covers several important manifold families.
  • Conclusion: The work establishes a unified geometric framework for extending proportional analogies beyond symbolic domains and Euclidean vector spaces.The framework is intended to support analogy formulations in non-Euclidean settings.
  • Conclusion: Future directions include non-Riemannian extensions, efficient numerical solvers on general manifolds, and empirical studies of analogy-based learning for transfer learning, meta-learning, and sparse coding.These directions are presented as areas for further exploration and potential applications.
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