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A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds
Pierre-Alexandre Murena, Marcelo Hartmann
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Summary
This paper introduces a generalized parallelogram rule for proportional analogies on Riemannian manifolds. It extends the concept of analogical proportions (a:b::c:d) from Euclidean vector spaces to non-Euclidean Riemannian domains by utilizing geodesic midpoints instead of parallel transport. The authors prove that this intrinsic definition satisfies the axioms of proportional analogies, including symmetry, exchange of means, and reflexivity. The framework is shown to be invariant under isometries and robust against noise on Hadamard manifolds. The method is illustrated on various manifolds, including symmetric positive definite matrices, Kendall's shape space, 3D triangular meshes, and the simplex of probability distributions.
Entities (10)
Relation Signals (9)
Generalized Parallelogram Rule → appliesto → Riemannian manifold
confidence 95% · In this paper, we introduce a proportional analogy relation in Riemannian domains, extending the parallelogram rule used for arithmetic analogies in Euclidean spaces.
Generalized Parallelogram Rule → satisfies → Proportional Analogy
confidence 95% · We introduce a new intrinsic definition of proportional analogies on Riemannian manifolds based on geodesic midpoints and prove that it satisfies the axioms of proportional analogies.
Generalized Parallelogram Rule → uses → Geodesic Midpoint
confidence 95% · Rather than characterizing analogies through parallel displacements, we characterize them through geodesic midpoints.
Generalized Parallelogram Rule → illustratedon → Symmetric Positive Definite Matrices
confidence 90% · Domain 1 corresponds to Symmetric Positive Definite Matrices, with results presented in Corollary 11.
Generalized Parallelogram Rule → illustratedon → Simplex of Probability Distributions
confidence 90% · Domain 4 is the simplex of probability distributions, endowed with the Aitchison metric
Generalized Parallelogram Rule → illustratedon → Kendall's Space
confidence 90% · Domain 2 is Kendall’s space of triangular shapes, with results presented in Corollary 8.
Proportional Analogy → isformalizedby → Parallelogram Rule
confidence 90% · In Euclidean spaces, the classical parallelogram can equivalently be described by the fact that the diagonals share the same midpoint.
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Abstract
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.
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- Source: https://arxiv.org/abs/2608.14220v1
- Canonical: https://arxiv.org/abs/2608.14220v1
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A Generalized Parallelogram Rule for Proportional Analogies on Riemannian Manifolds Pierre-Alexandre Murena Marcelo Hartmann Abstract Analogies are quaternary relations of the form “a is to b as c is to d”, usually denoted a:b::c:da: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. Code — https://github.com/ppaamm/RiemannianAnalogies Introduction 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. Analogies are quaternary relations of the form “a is to b as c is to d”, usually written a:b::c:da:b::c:d (19). The capability to understand and solve analogies, i.e. to find the d that makes it correct, is recognized as one of the core characteristics of human cognition (14). It has also been used as a validation tool for machine learning methods. An early notable case was the use of analogies to demonstrate the validity of word embeddings such as word2vec (13): although these models were not explicitly trained to learn analogies, it was observed that the parallelogram rule b−a=d−cb-a=d-c (22) applied in the learned latent space yielded valid analogies. While this finding has been widely debated (18), it nonetheless helped establish analogies as a tool for validation. More recently, analogies have been used to evaluate the reasoning abilities of Large Language Models (LLMs) (26), and the ARC-AGI challenge considers analogy-making as a central capability of Artificial General Intelligence (5). Various models of analogies have been proposed, among which proportional analogies have been particularly popular (11). This model is an axiomatic characterization, stating that a:b::c:da:b::c:d holds on some conditions of the four terms. A strength of this model is that it is consistent with the philosophical roots of analogies while applying to a large number of domains (19). In particular, the parallelogram rule in vector spaces is a valid proportional analogy. However, most of the research was conducted either on symbolic domains (character strings, Boolean values, logic formulae) or in vector spaces, ignoring more complex structures such as manifolds. The preliminary work of 15 is an attempt to extend the parallelogram rule in Riemannian manifolds, but finds two limitations: the direct generalization of the rule, based on geodesic shooting and parallel transport, does not produce a valid proportional analogy, and the general construction of proportional analogies does not guarantee continuity properties. In this paper, we propose a different perspective. Rather than characterizing analogies through parallel displacements, we characterize them through geodesic midpoints. In Euclidean spaces, the classical parallelogram can equivalently be described by the fact that the diagonals share the same midpoint. We show that this characterization extends naturally to arbitrary Riemannian manifolds by replacing Euclidean midpoints with intrinsic geodesic midpoints. The resulting construction is entirely intrinsic, requires only exponential and logarithmic maps and satisfies the axioms of proportional analogies. Our contributions are fourfold: • We introduce a new intrinsic definition of proportional analogies on Riemannian manifolds based on geodesic midpoints and prove that it satisfies the axioms of proportional analogies. • We establish several theoretical properties of the proposed framework, including invariance under Riemannian isometries, robustness, and explicit characterizations on geodesically convex and symmetric spaces. • We extend the results of 10, showing that any quadruple of pairwise distinct points of ℝnR^n can be made analogous for a suitable metric when n≥2n≥ 2, and proposing ordering conditions for n=1n=1. • We illustrate the versatility of the framework on diverse geometric domains (Figure 1), demonstrating that a single intrinsic construction naturally extends analogical reasoning far beyond Euclidean vector spaces. Proportional Analogies An analogy is a quaternary relation, written “A:B::C:DA:B::C:D" and which reads “A is to B as C is to D". A common characterization of analogies, inspired by the philosophical works on proportions of Plato and Aristotle, relies on the satisfaction of three properties: Definition 1 (Proportional analogy). A quaternary relation of the form a:b::c:da:b::c:d is a proportional analogy if it satisfies the following properties for all a,b,c,da,b,c,d: • a:b::c:d⇔c:d::a:ba:b::c:d c:d::a:b (symmetry) • a:b::c:d⇔a:c::b:da:b::c:d a:c::b:d (exchange of the means) • a:b::a:ba:b::a:b (reflexivity) A proportional analogy is called strong if it satisfies a:b::a:x⇒x=ba:b::a:x x=b for all a and b. From the first two postulates, 8 equivalent forms can be derived: Proposition 1. For a proportional analogy, the following forms are equivalent for any a,b,ca,b,c and d: a:b::c:da:c::b:db:d::a:c a:b::c:d a:c::b:d b:d::a:c b:a::d:cd:c::b:ad:b::c:a b:a::d:c d:c::b:a d:b::c:a c:a::d:bc:d::a:b c:a::d:b c:d::a:b In a context of vector spaces, it is easy to verify that the parallelogram rule (22) corresponds to a proportional analogy. Proposition 2. The relation a:b::Ec:da:b::_Ec:d on a,b,c,da,b,c,d in a vector space E holding true when b−a=d−cb-a=d-c is a proportional analogy, called arithmetic proportion. Various interpretations can be given of this rule, for instance that b differs from a the same way d differs from c. Equivalent interpretations can be given by on the equivalent forms presented in Proposition 1. These interpretations all consist in comparing two by two the parallel sides of the parallelogram. An alternative vision presented by 10 takes a different perspective on the parallelogram: instead of characterizing the parallelogram by its parallel sides, they propose to describe it in terms of its center: four points a,b,ca,b,c and d are in analogy if and only if the middle of the segment [a,c][a,c] coincides with the middle of the segment [b,d][b,d]. The authors use the notion of generalized mean (8) to propose a family of analogical proportions. Proposition 3. Let mpm_p be a generalized mean of order p∈ℝp defined as mp(a,b)=limr→par+br2rm_p(a,b)= _r→ p [r] a^r+b^r2. Then the relation a:b::p:c:da:b::_p:c:d which holds true if and only if mp(a,c)=mp(b,d)m_p(a,c)=m_p(b,d) is a proportional analogy. An important question is to solve analogical equations, i.e. to find x such that a:b::c:xa:b::c:x for a given proportional analogy. In the following, we use the notation SolX(a,b,c)=d∈X|a:b::c:dSol_X(a,b,c)=\d∈ X\,|\,a:b::c:d\ (1) to designate the set of solutions of the proportional analogy in the space X. Riemannian Geometry A topological manifold of dimension d is defined as a paracompact Haussdorf space in which every point has a neighborhood U that is homeomorphic to an open subset of ℝdR^d, called chart. A topological manifold is called differentiable if the mapping from one chart to the other is smooth. A tangent vector ξ to a differentiable manifold ℳM at point p is the equivalence class of the differentiable curves γ on ℳM such that γ(0)=pγ(0)=p modulo a first-order contact condition between the curves. The set of all tangent vectors to a point p∈ℳp is denoted TpℳT_pM and can be shown to be a d-dimensional vector space. The tangent bundle TℳTM is defined as the disjoint union of all tangent vectors: Tℳ=⨆p∈ℳTpℳTM= _p T_pM The tangent space TpℳT_pM can be equipped with an inner product gp:(Tpℳ)2→ℝg_p:(T_pM)^2 . When gpg_p varies smoothly with p and it is positive, we call (ℳ,g)(M,g) a Riemannian manifold. A connection ∇ on a smooth manifold ℳM is a bilinear map ∞(Tℳ)×∞(Tℳ)→∞(Tℳ)C^∞(TM)×C^∞(TM) ^∞(TM), written ∇XY _XY, such that ∇fXY=f∇XY _fXY=f\, _XY and ∇X(fY)=X[f]Y+f∇XY _X(fY)=X[f]Y+f\, _XY for all X,Y∈TℳX,Y∈ TM and all f∈∞(ℳ)f ^∞(M). Definition 2. Let (ℳ,g)(M,g) be a Riemannian manifold and let ∇ be a connection on ℳM. A smooth curve γ:[0,1]→ℳγ:[0,1] is said to be a geodesic if ∇γ˙γ˙=0 _ γ γ=0. It can be verified that, for all p∈ℳp and V∈TpℳV∈ T_pM, there exists a unique geodesic such that γ(0)=pγ(0)=p and γ˙(0)=V γ(0)=V. Definition 3 (Exponential map). Let (ℳ,g)(M,g) be a Riemannian manifold and let ∇ be a connection on ℳM. The exponential map at a point p∈ℳp is the map expp:p→ℳ _p:D_p , with p⊆TpℳD_p T_pM such that for all V∈Tpℳ,expp(V)=γ(1)V∈ T_pM, _p(V)=γ(1) where γ is the unique geodesic such that γ(0)=pγ(0)=p and γ˙(0)=V γ(0)=V. When p=TpℳD_p=T_pM of all p, we say that ℳM is geodesically complete. Proposition 4. Under the same conditions as the previous definition, the path t↦expp(tV)t _p(tV) is the geodesic starting at p and with initial velocity V. A Riemannian manifold (ℳ,g)(M,g) is said to be geodesically convex when there exists a unique geodesic between any two points on ℳM. In this case, we define the logarithmic map, denoted by logp(q) _p(q), as the unique vector V∈TpℳV∈ T_pM such that q=expp(V)q= _p(V). Definition 4 (Arclength). Let γ:[a,b]→ℳγ:[a,b] be a smooth curve on the Riemannian manifold (ℳ,g)(M,g). The arclength of γ from t1t_1 to t2t_2 (t1≤t2)(t_1≤ t_2) is defined as: L(γ)=∫t1t2‖γ˙(t)‖tL(γ)= _t_1^t_2\| γ(t)\|\,dt (2) Proposition 5. Let (ℳ,g)(M,g) be a Riemannian manifold and let ∇ be a connection on ℳM. Given p∈ℳp and V∈TpℳV∈ T_pM, let γτ:[0,1]→ℳ _τ:[0,1] be the geodesic defined by γτ(t)=expp(tτV) _τ(t)= _p(tτ V) for t∈[0,1]t∈[0,1]. Then L(γτ)=τL(γ1)L( _τ)=τ L( _1). Proof. Since γτ _τ is a geodesic, the quantity ‖γ˙τ(t)‖g\| γ_τ(t)\|_g is constant, and equal to ‖τV‖g\|τ V\|_g. Then L(γτ)=gp(τV,τV)=τgp(V,V)=τL(γ1)L( _τ)= g_p(τ V,τ V)=τ g_p(V,V)=τ L( _1). ∎ Proportional Analogies based on the Generalized Parallelogram Proportional Analogies on Geodesically Convex Riemannian Manifolds The construction process of the parallelogram used in arithmetic analogies on vector spaces can be naturally extended to Riemannian manifolds (15). Instead of considering the vectors b−ab-a and d−cd-c, we consider the logarithmic map loga(b) _a(b), defined as a the vector V∈TaℳV∈ T_aM such that expa(V)=b _a(V)=b. Given a,ba,b and c, a natural solution to compute d is to compute loga(b) _a(b), to transport it to c and to shoot the corresponding geodesic from c. However, such a process does not result in a proportional analogy, since the d obtained by solving a:b::c:xa:b::c:x and a:c::b:xa:c::b:x is not the same. This is a consequence of the curvature of the manifold. aabbccddmmγa,d _a,dγb,c _b,c Figure 2: Analogy a:b::c:da:b::c:d holds since the geodesics γa,d _a,d from a to d and γb,c _b,c from b to c intersect at their midpoint. In this paper, we propose an alternative characterization, following the idea of 10. Proposition 6. Let (ℳ,g)(M,g) be a geodesically convex Riemannian manifold and let ∇ be a connection on ℳM. The following relation is a strong proportional analogy on ℳM. We have a:b::c:da:b::c:d if and only if expa(12loga(d))=expb(12logb(c)) _a ( 12 _a(d) )= _b ( 12 _b(c) ) (3) The proof is trivial given the following lemma: Lemma 1. Given p,q∈ℳp,q , we have expp(12logp(q))=expq(12logq(p)) _p ( 12 _p(q) )= _q ( 12 _q(p) ) Proof. We denote by γ the geodesic from p to q defined as t↦expp(tlogp(q))t _p(t _p(q)). Consider the geodesic in the reverse direction: γ~(t)=γ(1−t) γ(t)=γ(1-t). Since γ~˙(t)=−γ˙(1−t) γ(t)=- γ(1-t), we have for all t: 0=∇γ~˙(t)γ~˙(t)=∇−γ˙(1−t)(−γ˙(1−t))=∇γ˙(1−t)(γ˙(1−t))0= _ γ(t) γ(t)= _- γ(1-t)(- γ(1-t))= _ γ(1-t)( γ(1-t)) which shows that γ~ γ is a geodesic. Since γ~(0)=q γ(0)=q and γ~(1)=p γ(1)=p, we can conclude that γ~˙(0)=logq(p) γ(0)= _q(p) and that γ~(t)=expq(tlogq(p)) γ(t)= _q(t _q(p)). expp(12logp(q)) _p ( 12 _p(q) ) =γ(12) =γ ( 12 ) =γ~(12)=expq(12logq(p)) = γ ( 12 )= _q ( 12 _q(p) ) which proves the result. ∎ Proportional Analogies on Riemannian Manifolds In the general case, there is no unicity of a geodesic and the logarithmic map is only a local operator (i.e. logp(q) _p(q) is defined for q in a neighborhood of p). A standard example is the 2-dimensional sphere 2S^2 on which there exist infinitely many geodesics between the two poles. We now propose a generalization of the proportional analogy defined in Equation 3. Proposition 7. Let (ℳ,g)(M,g) be a Riemannian manifold and let ∇ be a connection on ℳM. The following relation is a proportional analogy on ℳM. We have a:b::c:da:b::c:d if and only if: ∃v∈Taℳ,∃w∈Tbℳ;c=expb(w)d=expa(v)expa(v2)=expb(w2)∃ v∈ T_aM,\,∃ w∈ T_bM;\, casesc= _b(w)\\ d= _a(v)\\ _a ( v2 )= _b ( w2 ) cases (4) In the general case, the analogy defined in Equation 4 is not a strong analogy: given a and b, we have a:b::a:ba:b::a:b but there exists some x≠bx≠ b such that a:b::a:xa:b::a:x. Corollary 1. When (ℳ,g)(M,g) is geodesically convex, the proportional analogies defined in Equation 3 and in Equation 4 are the same. This corollary requires that the whole manifold is geodesically convex, which is a strong assumption. We have a weaker result, stating the two proportional analogies coincide only for points where the condition on the unicity of the geodesic is satisfied. Corollary 2. Let a,b,c,d∈ℳa,b,c,d . If there exists a unique v∈Taℳv∈ T_aM and a unique w∈Tbℳw∈ T_bM such that c=expb(w)c= _b(w) and d=expa(v)d= _a(v), then a:b::c:da:b::c:d if and only if expa(v2)=expb(w2) _a ( v2 )= _b ( w2 ). From the way we defined the analogy, we can derive a way to solve analogical equations, i.e. to find the values of x such that a:b::c:xa:b::c:x. Corollary 3. Let a,b,c∈ℳa,b,c . If there exists a unique v∈Taℳv∈ T_aM and a unique w∈Tbℳw∈ T_bM such that c=expb(w)c= _b(w) and expa(v2)=expb(w2) _a ( v2 )= _b ( w2 ), then a:b::c:xa:b::c:x if and only if x=expa(v)x= _a(v). This corresponds to Solℳ(a,b,c)=expa(v)Sol_M(a,b,c)=\ _a(v)\. Isometry Invariance Proposition 8 (Invariance under isometries). Let (ℳ,g)(M,g) and (,h)(N,h) be Riemannian manifolds, and let ϕ:M→Nφ:M→ N be a Riemannian isometry. If a:b::(ℳ,g)c:da:b::_(M,g)c:d holds according to the proportional analogy defined in Equation 4, then ϕ(a):ϕ(b)::(,h)ϕ(c):ϕ(d)φ(a):φ(b)::_(N,h)φ(c):φ(d) also holds. Proof. Assume that a:b::(ℳ,g)c:da:b::_(M,g)c:d. There exist tangent vectors v∈Taℳv∈ T_aM and w∈Tbℳw∈ T_bM such that d=expa(v)d= _a(v), c=expb(w)c= _b(w), and the midpoint reached from a along v is the same as the midpoint reached from b along w. The geodesic γv(t)=expa(tv) _v(t)= _a(tv) is such that γv(0)=a _v(0)=a and γv˙(0)=v _v(0)=v. ϕφ being an isometry, it preserves the geodesics, and consequently the curve γv~(t)=ϕ(γv(t)) _v(t)=φ( _v(t)) is also a geodesic such that γv~(0)=ϕ(a) _v(0)=φ(a) and γv~˙(0)=dϕa(v) _v(0)=d _a(v). Consequently, for every t,ϕ(expa(tv))=expϕ(a)(tdϕa(v))t,φ( _a(tv))= _φ(a)(t\,d _a(v)). The same result holds for γw(t)=expb(tw) _w(t)= _b(tw). Consequently, we have ϕ(d)=expϕ(a)(dϕa(v))φ(d)= _φ(a)(d _a(v)), ϕ(c)=expϕ(b)(dϕb(w))φ(c)= _φ(b)(d _b(w)) and expϕ(a)(12dϕa(v))=expϕ(b)(12dϕb(w)) _φ(a)( 12d _a(v))= _φ(b)( 12d _b(w)), which proves that ϕ(a):ϕ(b)::(,h)ϕ(c):ϕ(d)φ(a):φ(b)::_(N,h)φ(c):φ(d). ∎ Robust Proportional Analogies As mentioned in the introduction, most previous works on proportional analogies focus on Boolean and symbolic domains. When working on continuous domains, we may need the additional continuity property, indicating that the analogical relation can be preserved under small perturbations. In the context of a relation, this continuity property is assessed by the notion of robustness, which interprets as the resistance against noise. Definition 5 (Robust proportional analogy). Let (X,Δ)(X, ) be a metric space. A proportional analogy on X is called robust in (a,b,c,d)∈X4(a,b,c,d)∈ X^4 if, for all ε>0 >0, there exists (a′,b′,c′,d′)≠(a,b,c,d)(a ,b ,c ,d )≠(a,b,c,d) such that Δ(a,a′)<ε (a,a )< , Δ(b,b′)<ε (b,b )< , Δ(c,c′)<ε (c,c )< and Δ(d,d′)<ε (d,d )< , and a′:b′::c′:d′a :b ::c :d . A constructive approach to analogies consists in building d given a,ba,b and c, ensuring that a:b::c:da:b::c:d. For instance, the arithmetic proportion (Proposition 2) can be seen as constructing d=c+b−ad=c+b-a. Definition 6 (Function-based proportional analogy). A proportional analogy is called function-based if there exists a function f:X3→Xf:X^3→ X such that a:b::c:da:b::c:d if and only if d=f(a,b,c)d=f(a,b,c). Proposition 9. Let (ℳ,g)(M,g) be a complete, uniquely geodesic Riemannian manifold. Then the analogy defined in Proposition 6 is a function-based proportional analogy. Proof. We simply take f(a,b,c)=expa(2loga(expb(12logb(c))))f(a,b,c)= _a (2 _a ( _b ( 12 _b(c) ) ) ) (5) ∎ Proposition 10. Let f:X3→Xf:X^3→ X define a function-based proportional analogy. If f is continuous at (a,b,c)(a,b,c) and (a,b,c)(a,b,c) is not isolated in X3X^3, then the analogy is robust in (a,b,c,f(a,b,c))(a,b,c,f(a,b,c)). Proof. Assume that f is continuous in (a,b,c)(a,b,c). Let ε>0 >0. There exists δa,δb,δc>0 _a, _b, _c>0 such that if Δ(a,a′)<δa (a,a )< _a, Δ(b,b′)<δb (b,b )< _b and Δ(c,c′)<δc (c,c )< _c, then Δ(f(a,b,c),f(a′,b′,c′))<ε (f(a,b,c),f(a ,b ,c ))< . In particular, we can find a′a such that Δ(a,a′)<minδa,ε≤ε (a,a )< \ _a, \≤ , b′b such that Δ(b,b′)<minδb,ε≤ε (b,b )< \ _b, \≤ and c′c such that Δ(c,c′)<minδc,ε≤ε (c,c )< \ _c, \≤ guaranteeing that d′=f(a′,b′,c′)d =f(a ,b ,c ) is such that Δ(d,d′)<ε (d,d )< . By definition of function-based analogies, we have a′:b′::c′:d′a :b ::c :d , which shows the robustness. ∎ Combining the results of Propositions 9 and 10, and since the exp maps is a diffeomorphism on a Hadamard manifold, we obtain the following important result: Corollary 4. On a Hadamard manifold of positive dimension, the analogy of Proposition 6 is robust. Proportional Analogies on ℝnR^n We begin by investigating analogies in the Euclidean space ℝnR^n, where the geometry remains sufficiently simple to derive several general structural results. In particular, this setting allows us to characterize the realizability of analogies and highlights the role of the underlying Riemannian metric. Analogies on ℝR We first consider the simple case where ℳ=ℝM=R. In that case, the midpoint has a very simple characterization: Lemma 2. For any x0∈ℝx_0 , the function ϕ:x↦∫x0xg(t)tφ:x _x_0^x g(t)\,dt is a diffeomorphism, and the midpoint between x and y with metric g is: mg(x,y)=ϕ−1(ϕ(x)+ϕ(y)2).m_g(x,y)=φ^-1 ( φ(x)+φ(y)2 ). (6) Proof. ϕφ is differentiable with ϕ′(x)>0φ (x)>0 for all x, so ϕφ is a strictly increasing diffeomorphism from ℝR to its image. Assuming x<yx<y, the midpoint m=mg(x,y)m=m_g(x,y) is defined by: ∫xmg(t)t=∫myg(t)t _x^m g(t)\,dt= _m^y g(t)\,dt which is equivalent to ϕ(m)−ϕ(x)=ϕ(y)−ϕ(m)φ(m)-φ(x)=φ(y)-φ(m). Since the image ϕ(ℝ)φ(R) of ϕφ is a segment, 12(ϕ(x)+ϕ(y))∈ϕ(ℝ) 12(φ(x)+φ(y))∈φ(R), and m=ϕ−1(ϕ(x)+ϕ(y)2).m=φ^-1 ( φ(x)+φ(y)2 ). ∎ Theorem 1. Let a,b,c,d∈ℝa,b,c,d be pairwise distinct points, and Ix,y=(minx,y,maxx,y)I_x,y=( \x,y\, \x,y\) the open interval between x and y. There exists a smooth Riemannian metric g on ℝR such that the analogy a:b::gc:da:b::_gc:d holds, if and only if Ib,c⊂Ia,dI_b,c⊂ I_a,d or Ia,d⊂Ib,cI_a,d⊂ I_b,c. Proof. Let x1<x2<x3<x4x_1<x_2<x_3<x_4 be the four points a,b,c,da,b,c,d arranged in increasing order. We first prove necessity. Suppose that a:b::gc:da:b::_gc:d but that the intervals Ia,dI_a,d and Ib,cI_b,c are not nested. There are two possible pairings, up to exchanging the two pairs. The first possibility is a,d=x1,x2\a,d\=\x_1,x_2\ and b,c=x3,x4\b,c\=\x_3,x_4\. Since ϕφ is strictly increasing, ϕ(x1)+ϕ(x2)<ϕ(x3)+ϕ(x4)φ(x_1)+φ(x_2)<φ(x_3)+φ(x_4) and ϕ(a)+ϕ(d)≠ϕ(b)+ϕ(c).φ(a)+φ(d)≠φ(b)+φ(c). The second possibility is a,d=x1,x3\a,d\=\x_1,x_3\ and b,c=x2,x4\b,c\=\x_2,x_4\. Again, strict monotonicity gives ϕ(x1)<ϕ(x2)φ(x_1)<φ(x_2) and ϕ(x3)<ϕ(x4)φ(x_3)<φ(x_4). Adding these inequalities yields ϕ(x1)+ϕ(x3)<ϕ(x2)+ϕ(x4)φ(x_1)+φ(x_3)<φ(x_2)+φ(x_4), and hence ϕ(a)+ϕ(d)≠ϕ(b)+ϕ(c)φ(a)+φ(d)≠φ(b)+φ(c). Thus, if the two intervals are not nested, no smooth Riemannian metric can satisfy a:b::gc:da:b::_gc:d. We now prove sufficiency. Suppose that the intervals are nested. Since the four points are pairwise distinct, the larger interval must have endpoints x1x_1 and x4x_4, while the smaller interval has endpoints x2x_2 and x3x_3. Therefore, a,d=x1,x4\a,d\=\x_1,x_4\ and b,c=x2,x3\b,c\=\x_2,x_3\ or conversely. It is enough to construct a smooth positive function ρ such that ∫x1x2ρ(t)t=∫x3x4ρ(t)t _x_1^x_2ρ(t)\,dt= _x_3^x_4ρ(t)\,dt. Suppose first that x2−x1≤x4−x3x_2-x_1≤ x_4-x_3. Let ψ∈Cc∞((x1,x2))ψ∈ C_c^∞((x_1,x_2)) be a non-negative smooth function such that ∫x1x2ψ(t)t=1 _x_1^x_2ψ(t)\,dt=1 and ψ(t)=0ψ(t)=0 outside [x1,x2][x_1,x_2]. We define ρ(t)=1+((x4−x3)−(x2−x1))ψ(t)ρ(t)=1+((x_4-x_3)-(x_2-x_1))ψ(t). We can check that ∫x1x2ρ(t)t=∫x3x4ρ(t)t _x_1^x_2ρ(t)\,dt= _x_3^x_4ρ(t)\,dt. We can build a function ρ similarly in the case where x2−x1>x4−x3x_2-x_1>x_4-x_3. In both cases, ρ is smooth, strictly positive, and can therefore be used to define the metric g(t)=ρ(t)2g(t)=ρ(t)^2 satisfying mg(a,d)=mg(b,c)m_g(a,d)=m_g(b,c). ∎ The generalized means considered by 10 can be interpreted as Riemannian midpoints on the positive real line. Indeed, the power mean of order p≠0p≠ 0 is induced by the metric gp(x)=p2x2p−2g_p(x)=p^2x^2p-2, while the geometric mean is induce by the metric g0(x)=x−2g_0(x)=x^-2. Their existence and uniqueness theorem for 0<a<b<c<d0<a<b<c<d therefore establishes that, for every nested quadruple of positive points, there exists a unique metric within this particular one-parameter family that realizes the analogy. Our result characterizes existence over the full class of smooth Riemannian metrics and on the whole real line, and shows that nestedness is also necessary. Analogies on ℝnR^n for n≥2n≥ 2 The case n=1n=1 presented the difficulty that the intervals had to be nested in order to have the correct properties of a metric. In the general case, there is no global order anymore, which removes the constraints on the geodesics. Theorem 2. Let a,b,c,d∈ℝna,b,c,d ^n pairwise distinct points of ℝnR^n, with n≥2n≥ 2. There exists a metric g such that a:b::gc:da:b::_gc:d. Proof. Consider four pairwise distinct points A,B,C,D∈ℝnA,B,C,D ^n such that A+D=B+CA+D=B+C. The group Diff(ℝn)Diff(R^n) acts k-transitively on ℝnR^n for every finite k when n≥2n≥ 2 (12), and in particular acts 44-transitively. Consequently, there exists a diffeomorphism ϕ:ℝn→ℝnφ:R^n ^n such that ϕ(a)=Aφ(a)=A, ϕ(b)=Bφ(b)=B, ϕ(c)=Cφ(c)=C and ϕ(d)=Dφ(d)=D. Let gEg_E be the Euclidean metric on ℝnR^n. The pullback metric g=ϕ∗gEg=φ^*g_E on ℝnR^n is defined as gx(u,v)=gE,ϕ(x)(Dϕx(u),Dϕx(v))g_x(u,v)=g_E,φ(x)(D _x(u),D _x(v)). The function ϕφ is then an isometry from (ℝn,g)(R^n,g) onto (ℝn,gE)(R^n,g_E). Since A:B::gEC:DA:B::_g_EC:D, Proposition 8 guarantees that a:b::gc:da:b::_gc:d. ∎ The intrinsic difference between the cases n=1n=1 and n≥2n≥ 2 lies in the possibility to build a diffeomorphism, indicating that the realizability of analogies depends on the transitivity properties of the diffeomorphism group of the underlying manifold. Proportional Analogies on Riemannian Symmetric Spaces Riemannian symmetric spaces, introduced by 3, constitute one of the central classes of Riemannian manifolds and play a fundamental role in differential geometry and Lie theory. They include Euclidean spaces, spheres, hyperbolic spaces, Grassmann manifolds and the manifold of symmetric positive-definite matrices (7). Definition 7 (Riemannian symmetric space). A connected Riemannian manifold (ℳ,g)(M,g) is symmetric if for every point p∈ℳp , there exists an isometry sp:ℳ→ℳs_p:M such that sp(p)=ps_p(p)=p and D(sp)p=−IdD(s_p)_p=-Id. From this definition, it follows that, at every point p of a Riemannian symmetric space, any geodesic will be reflected, in the sense that γ(t)γ(t) is transformed into γ(−t)γ(-t). This property induces an important characterization of analogies. Theorem 3. Let (ℳ,g)(M,g) be a Riemanian symmetric space. For all a,b,c,d∈ℳa,b,c,d , we have a:b::gc:da:b::_gc:d if and only if there exists m∈Mid(b,c)m (b,c) such that d=sm(a)d=s_m(a), where: Mid(b,c)=expb(w2):w∈TbM,expb(w)=c.Mid(b,c)= \ _b ( w2 ):w∈ T_bM, _b(w)=c \. (7) Proof. We first prove the forward implication. Assume that a:b::gc:da:b::_gc:d. Then there exists v∈Taℳv∈ T_aM and w∈Tbℳw∈ T_bM satisfying the conditions presented in Proposition 4. Define m=expa(v/2)=expb(w/2)m= _a(v/2)= _b(w/2). In particular, we observe that m∈Mid(b,c)m (b,c). Consider the geodesic γ(t)=expa(tv)γ(t)= _a(tv) for t∈ℝt . It satifies γ(0)=aγ(0)=a, γ(1/2)=mγ(1/2)=m and γ(1)=dγ(1)=d. We reparametrize this geodesic as γ~ γ defined by γ~(t)=γ(t+1/2) γ(t)=γ(t+1/2). By construction, γ~ γ is a geodesic through m such that γ~(0)=m γ(0)=m, γ~(−1/2)=a γ(-1/2)=a and γ~(1/2)=d γ(1/2)=d. By definition of the geodesic symmetry, we have sm(γ(t)~)=γ~(−t)s_m( γ(t))= γ(-t) for all t. In particular for t=−1/2t=-1/2, we obtain d=γ~(1/2)=sm(γ~(−1/2))=sm(a)d= γ(1/2)=s_m( γ(-1/2))=s_m(a). For the reverse implication, we consider m∈Mid(b,c)m (b,c) such that d=sm(a)d=s_m(a). Since a Riemannian symmetric space is geodesically complete, there exists v∈Tmℳv∈ T_mM such that a=expm(v)a= _m(v). By definition of the symmetry, we have sm(expm(v))=expm(−v)s_m( _m(v))= _m(-v). Hence d=sm(a)=expm(−v)d=s_m(a)= _m(-v). Consider the curve γ(t)=expm((1−2t)v)γ(t)= _m((1-2t)v) for t∈ℝt . It is an affinely parametrized geodesic, and it satisfies γ(0)=aγ(0)=a, γ(1/2)=mγ(1/2)=m and γ(1)=dγ(1)=d. Denoting u=γ˙(0)∈Taℳu= γ(0)∈ T_aM, γ is the geodesic starting at a and with initial velocity u, so d=expa(u)d= _a(u) and m=expa(u/2)m= _a(u/2). Since m=expb(w/2)m= _b(w/2) and c=expb(w)c= _b(w), we have a:b::gc:da:b::_gc:d by Proposition 4. ∎ A direct application of this result is the Euclidean case. We can observe that (ℝn,gE)(R^n,g_E) is a Riemanian symmetric space with sm(a)=2m−as_m(a)=2m-a. When m is the midpoint between b and c, we have m=b+c2m= b+c2, and we retrieve the parallelogram rule d=c+b−ad=c+b-a. Proposition 11 (Robustness on Riemannian symmetric spaces). Let (ℳ,g)(M,g) be a Riemannian symmetric space of positive dimension. The proportional analogy defined in Theorem 7 is robust at every (a,b,c,d)∈ℳ4(a,b,c,d) ^4 such that a:b::(ℳ,g)c:da:b::_(M,g)c:d: Proof. Assume that a:b::(ℳ,g)c:da:b::_(M,g)c:d. By Theorem 7, there exists m∈Mid(b,c)m (b,c) such that d=sm(a)d=s_m(a). Let ε>0 >0. Since ℳM has positive dimension, there exists a′≠a ≠ a such that dist(a,a′)<εdist(a,a )< . Set b′=b,c′=c,d′=sm(a′).b =b, c =c, d =s_m(a ). Since m∈Mid(b′,c′)m (b ,c ), Theorem 7 implies a′:b′::(ℳ,g)c′:d′a :b ::_(M,g)c :d . Moreover, sms_m is a Riemannian isometry, and hence dist(d,d′)=dist(sm(a),sm(a′))=dist(a,a′)<ε.dist(d,d )=dist (s_m(a),s_m(a ) )=dist(a,a )< . The other two distances are zero: dist(b,b′)=dist(c,c′)=0.dist(b,b )=dist(c,c )=0. Finally, (a′,b′,c′,d′)≠(a,b,c,d)(a ,b ,c ,d )≠(a,b,c,d) because a′≠a ≠ a. Therefore, the analogy is robust at (a,b,c,d)(a,b,c,d). ∎ This result shows that valid analogical quadruples are not isolated: every neighborhood of a valid quadruple contains another valid quadruple. Corollary 4 does not provide a strong enough characterization here: the proposition states that the space does not need to be a Hadamard manifold to ensure robustness of the analogy when it is symmetric. Analogies on Spheres The sphere n⊂ℝn+1S^n ^n+1 is defined as the set n=x∈ℝn+1;‖x‖E2=1S^n=\x ^n+1\,;\,\|x\|_E^2=1\. It can be verified that the sphere nS^n is a Riemannian symmetric space, with sm(x)=2⟨m,x⟩m−xs_m(x)=2 m,x m-x, where ⟨.,.⟩ .,. designates the Euclidean scalar product in the ambient space ℝn+1R^n+1 (7). Corollary 5. The solutions of the analogical equation a:b::c:xa:b::c:x on the sphere nS^n are: Soln=2⟨m,a⟩m−a:m∈Mid(b,c)Sol_S^n= \2 m,a m-a\,:\,m (b,c) \ (8) When b and c are not antipodal (i.e. when c≠−bc≠-b), the set of midpoints contains two elements: Mid(b,c)=b+c‖b+c‖,−b+c‖b+c‖Mid(b,c)= \ b+c\|b+c\|,- b+c\|b+c\| \. Both midpoints induce the same geodesic symmetry, and the analogical equation therefore admits the unique solution d=⟨a,b+c⟩1+⟨b,c⟩(b+c)−ad= a,b+c 1+ b,c (b+c)-a. When b=−cb=-c, the midpoint is not unique anymore: Mid(b,−b)=m∈n:⟨m,b⟩=0Mid(b,-b)=\m ^n\,:\, m,b =0\, and Soln=d∈n:⟨d,b⟩=−⟨a,b⟩Sol_S^n=\d ^n\,:\, d,b =- a,b \. In particular, in the case where n=2n=2, this corresponds to the points of azimuthal angle θ=π−θaθ=π- _a, with θa _a the azimuthal angle of a. The sphere RnS^n_R of radius R has symmetry sm(x)=2R2⟨m,x⟩m−xs_m(x)= 2R^2 m,x m-x. Corollary 6. The solutions of the analogical equation a:b::c:xa:b::c:x on the sphere nS^n are: SolRn=2R2⟨m,a⟩m−a:m∈Mid(b,c)Sol_S^n_R= \ 2R^2 m,a m-a\,:\,m (b,c) \ (9) Analogies on Hyperbolic Spaces Define the n-dimensional hyperboloid model as ℍn=x∈ℝn+1:⟨x,x⟩L=−1,x0>0H^n=\x ^n+1\,:\, x,x _L=-1,\,x_0>0\ where ⟨.,.⟩L .,. _L is the Lorentzian inner product defined as ⟨x,y⟩L=−x0y0+∑i=1nxiyi x,y _L=-x_0y_0+ _i=1^nx_iy_i. The Riemannian metric on ℍnH^n is induced from the Lorentzian metric. Since all standard models of hyperbolic space are mutually isometric, working on ℍnH^n entails no loss of generality. We refer the reader to 21 for more details on hyperbolic spaces. It can be verified that the hyperboloid ℍnH^n is a Riemannian symmetric space with sm(x)=−2⟨x,m⟩Lm−xs_m(x)=-2 x,m _Lm-x. Unlike the sphere, hyperbolic space is a Hadamard manifold. Consequently, any two points are joined by a unique geodesic, and therefore admit a unique geodesic midpoint. For any b,c∈ℍnb,c ^n, the unique midpoint is m=b+c2−2⟨b,c⟩Lm= b+c 2-2 b,c _L. Corollary 7. The analogical equation a:b::c:xa:b::c:x on the hyperboloid ℍnH^n admits a unique solution: Solℍn=−a+⟨a,b+c⟩L1−⟨b,c⟩L(b+c)Sol_H^n= \-a+ a,b+c _L1- b,c _L(b+c) \ (10) Analogies on Symmetric Positive Definite Matrices A matrix M is symmetric positive definite if and only if MT=M^T=M and xTMx>0x^TMx>0 for all x∈ℝn∖0x ^n \0\. We denote by SPDnSPD_n the space of symmetric positive definite matrices of size n. Symmetric positive definite matrices play an important role in various applications of machine learning such as metric learning (27) or sparse coding (4). The tangent space TΣSPDnT_ SPD_n is the space of symmetric matrices. An important family of metrics for SPDnSPD_n is the affine-invariant metrics of the form gΣAI(V,W)=αTr(Σ−1VΣ−1W)+βTr(Σ−1V)Tr(Σ−1W)g_ ^AI(V,W)=α Tr( ^-1V ^-1W)+β Tr( ^-1V)Tr( ^-1W), with α>0α>0 and α+nβ>0α+nβ>0. 25 show that the manifold (SPDn,gAI)(SPD_n,g^AI) is a Riemannian symmetric space with symmetry sΣ(Λ)=ΣΛ−1Σs_ ( )= ^-1 . The midpoints between B and C are characterized by C=sM(B)=MB−1MC=s_M(B)=MB^-1M. The equation has a unique solution M=B#C=B1/2(B−1/2CB−1/2)1/2B1/2M=B\#C=B^1/2(B^-1/2CB^-1/2)^1/2B^1/2. Corollary 8. The analogical equation A:B::C:XA:B::C:X on the space of symmetric positive definite matrices SPDnSPD_n admits a unique solution: SolSPDn=(B#C)A−1(B#C)Sol_SPD_n= \(B\#C)A^-1(B\#C) \ (11) Proportional Analogies on Shape Spaces In this section, we explore how the introduced analogies apply to shape spaces. We will start with the simple case of Kendall’s space of triangles (9), which reduces to the case of the sphere 2S^2, and will then extend to 3D shapes, with potential applications in computer graphics. A Short Introduction to Shape Spaces Many tasks in computer vision, medical imaging and computer graphics rely on comparing geometric objects rather than individual pixels. Examples include object recognition, motion analysis and statistical shape analysis. In these settings, the quantity of interest is the shape of an object, independently of transformations such as translation, rotation or scaling. A natural way to formalize this idea is to represent an object by a finite set of landmarks and to identify configurations that differ only by these nuisance transformations. The resulting quotient space, called a shape space, is endowed with a Riemannian structure, allowing distances, geodesics and statistical quantities such as means to be defined intrinsically (9). Shapes can also be modeled as continuous curves or surfaces (23). In this setting, a shape is represented by an embedding of a reference manifold into the ambient space, and equivalent parameterizations are identified through the action of the diffeomorphism group. The resulting quotient spaces inherit a rich differential geometric structure. Space of Triangles A landmark configuration consisting of three points in the plane can be interpreted as a triangle. Since the shape of a triangle should be independent of its position, orientation and size, the construction of 9 first removes translations by centering the landmarks, then normalizes the configuration to unit size, and finally identifies configurations that differ only by a planar rotation. Let (z1,z2,z3)∈ℂ3(z_1,z_2,z_3) ^3 denote the coordinates of the three vertices of the triangle, interpreted as complex numbers. Quotienting by the translation and scale invariance results in the two-dimensional preshape space ∈ℂ3:∑i=13zi=0,∥=1\z ^3\,:\, _i=1^3z_i=0,\|z\|=1\. Quotienting this space by the action of planar rotations z∼eiθz e^iθz identifies preshapes representing the same triangle. The resulting quotient is Kendall’s shape space of planar triangles, which is isometric to the sphere 2S^2. Consequently, the result established in Corollary 8 for the sphere directly applies to planar triangles. Three-Dimensional Shapes As an illustration of the generalization capacity of our method, we consider analogical reasoning on three-dimensional triangle meshes. This application is closely related to the classical problem of deformation transfer (24), where a deformation observed on one shape is transferred to another. Our objective is not to compete with specialized deformation transfer methods, but rather to demonstrate that our general analogy relation naturally extends to a challenging geometric domain without requiring an application-specific formulation. For the geometric operations on shape spaces, we rely on the Morphomatics library (2). The library represents triangular meshes through fundamental coordinates, providing a Riemannian manifold structure on which we directly apply the analogy solver of Proposition 9. Given three meshes A, B, and C, the missing mesh D is obtained by solving the corresponding analogical equation. The experiments are conducted on animal meshes derived from the SMALR model (28). Since all meshes are generated from the same template, they share an identical triangulation and establish a one-to-one correspondence between vertices. This common topology satisfies the assumptions of the fundamental-coordinate representation; otherwise, a preprocessing step would be required to establish vertex correspondences. Figure 1 (domain 3) provides a typical example. Meshes A and B represent the same dog in a reference pose and a transformed pose, while mesh C represents a cow in the reference pose. The computed mesh D transfers the deformation A→BA→ B to the cow while preserving its geometric characteristics. Although not competitive with dedicated deformation transfer methods, the successful transfer and the low execution time (approximately 2s2\,s) illustrate the ability of the proposed framework to generalize beyond the domains for which it was originally designed. Proportional Analogies on Categorical Distributions The space of n-dimensional categorical distributions is defined as the simplex: Δn−1=p∈(0,1)n∣pi>0and∑i=1npi=1 ^n-1= \p∈(0,1)^n p_i>0 _i=1^np_i=1 \ (12) Several geometries have been proposed on the simplex of categorical distributions. A comprehensive comparison is given by 16. In this section, we consider two Riemannian geometries: the Fisher-Rao metric (17) and the Aitchison metric (6). Other geometries, such as the Hilbert geometry, are not Riemannian and do not enter the scope of this paper. Future works should investigate how our framework can be extended to such geometries. Fisher-Rao Metric Information Geometry (17) introduces Fisher-Rao metric as a natural metric choice for the manifolds of parametric probability distributions. In the specific case of categorical distribution, the Fisher-Rao metric is defined as: gFR,P(v1,v2)=∑i=1nv1iv2ipi.g_FR,P(v_1,v_2)= _i=1^n v_1^iv_2^ip_i. (13) The square-root mapping ϕ:Δn−1→2,+n−1φ: ^n-1 _2,+^n-1 defined as ϕ(p)=(2p1,…,2pn)φ(p)=(2 p_1,…c,2 p_n) is an isometry from the Fisher-Rao simplex to the positive orthant of the sphere of radius 22, denoted 2,+n−1S_2,+^n-1. Using the result of Proposition 8, we can show the following result: Proposition 12. Let a,b,c∈Δn−1a,b,c∈ ^n-1. Define r=2⟨a,b+c⟩‖b+c‖2(b+c)−a.r=2 a, b+ c \| b+ c\|^2( b+ c)- a. (14) The analogical equation a:b::(Δn−1,gFR)c:xa:b::_( ^n-1,g_FR)c:x admits a solution if and only if ri>0r_i>0 for all i. When the condition holds, the solution is unique and is given by d=(r12,…,rn2)d=(r_1^2,…c,r_n^2). Proof. Let A=ϕ(a)A=φ(a), B=ϕ(b)B=φ(b) and C=ϕ(c)C=φ(c). Since B,C∈2,+n−1B,C _2,+^n-1, the two points cannot be antipodal. Therefore, they admit a single midpoint M=2b+c‖b+c‖M=2 b+ c\| b+ c\|. We observe that M∈2,+n−1M _2,+^n-1, so A and M are not antipodal. By Corollary 9, the solution of the analogical equation on 2n−1S_2^n-1 is D=2rD=2r. It is a solution on 2,+n−1S_2,+^n-1 if and only if each ri>0r_i>0. The solution on Δn−1 ^n-1 is obtained by applying ϕ−1φ^-1. ∎ The fact that all analogical equations do not necessarily have a solution with Fisher-Rao metric can be explained by the fact that (Δn−1,gFR)( ^n-1,g_FR) is not geodesically complete: for a given p∈Δn−1p∈ ^n-1 and u∈TpΔn−1u∈ T_p ^n-1, the function t↦expp(tu)t _p(tu) is not defined for all t∈ℝt . Consequently, geodesics may leave the manifold in finite time, preventing some geodesic symmetries from being defined. This explains why analogical equations are not always solvable. Aitchison Metric Another approach consists in finding a diffeomorphism ψ:Δn−1→ℝn−1ψ: ^n-1 ^n-1 to endow Δn−1 ^n-1 with a pullback of the Euclidean metric. The solution proposed by 1 consists in introducing the centered log ratio clr(p)=(logp1f(p),…,logpnf(p))clr(p)= ( p_1f(p),…c, p_nf(p) ) with f(p)=(∏i=1npi)1/nf(p)= ( _i=1^np_i )^1/n the geometric mean of the vector p. The image of the clr mapping is the hyperplane H=x∈ℝn:∑i=1nxi=0H= \x ^n\,:\, _i=1^nx_i=0 \, and therefore defines a diffeomorphism ψ:Δn−1→ℝn−1ψ: ^n-1 ^n-1. We define gA=ψ∗gEg_A=ψ^*g_E as the pullback of the Euclidean metric gEg_E on ℝn−1R^n-1. Consequently, (Δn−1,gA)( ^n-1,g_A) is isometric to (ℝn−1,gE)(R^n-1,g_E), and we can apply Proposition 8. Proposition 13. Let a,b,c∈Δn−1a,b,c∈ ^n-1. The analogical equation a:b::(Δn−1,gA)c:xa:b::_( ^n-1,g_A)c:x has a unique solution: d=(c1b1/a1∑i=1ncibi/ai,…,cnbn/an∑i=1ncibi/ai)d= ( c_1b_1/a_1 _i=1^nc_ib_i/a_i,…c, c_nb_n/a_n _i=1^nc_ib_i/a_i ) (15) Proof. It follows directly from Proposition 8 that the analogy a:b::(Δn−1,gA)c:da:b::_( ^n-1,g_A)c:d holds if and only if clr(a):clr(b)::(H,gE)clr(c):clr(d)clr(a):clr(b)::_(H,g_E)clr(c):clr(d). Consequently, the equation has a unique solution satisfying for each component i: di=exp(clr(c)i+clr(b)i−clr(a)i)∑j=1nexp(clr(c)j+clr(b)j−clr(a)j)d_i= (clr(c)_i+clr(b)_i-clr(a)_i) _j=1^n (clr(c)_j+clr(b)_j-clr(a)_j) which can be developed into the solution of Equation 15. ∎ Application To evaluate the proposed analogical framework on categorical distributions, we consider a preference transfer task based on the MovieLens 1M dataset. Task description. For each demographic cohort g (e.g. age, gender, occupation) and movie category c present in the dataset, we compute the empirical distribution of ratings pgc=(pgc(1),…,pgc(5))p_gc=(p_gc(1),…c,p_gc(5)) where pgc(r)p_gc(r) is the probability that a member of the cohort g gives rating r to a movie of category c. To avoid zero probabilities, each distribution is estimated using symmetric Dirichlet smoothing. Figure 3: An example of a rating profile transfer between two cohorts (25−3425-34 men and <25<25 women) and two categories (action and film noir). A rating profile is represented as a distribution p∈Δ4p∈ ^4. The objective is to predict the rating distribution of a target cohort on a target category from three observed distributions. More precisely, given a reference cohort g1g_1, a target cohort g2g_2, a reference category c1c_1 and a target category c2c_2, we consider the analogical equation pg1,c1:pg1,c2::pg2,c1:xp_g_1,c_1:p_g_1,c_2::p_g_2,c_1:x whose solution is used as an estimate of the unknown distribution pg2,c2p_g_2,c_2 (Figure 3). Intuitively, the transformation induced by changing from category c1c_1 to c2c_2 for the reference cohort is transferred to the target cohort. Table 1: Mean Jensen-Shannon divergence (×10−3× 10^-3) on the MovieLens preference transfer task. Lower is better. Values in brackets denote 95% bootstrap confidence intervals. Experiment LOO Genre LOO Add. Fisher-Rao Aitchison Prade Age 1.323[1.108, 1.560]1.323_[1.108,\,1.560] 0.366[0.259, 0.487]0.366_[0.259,\,0.487] 0.274[0.190, 0.373]0.274_[0.190,\,0.373] 0.314[0.210, 0.451]0.314_[0.210,\,0.451] 0.294[0.208, 0.393]0.294_[0.208,\,0.393] Age + Gender 1.731[1.449, 2.027]1.731_[1.449,\,2.027] 0.842[0.641, 1.064]0.842_[0.641,\,1.064] 0.679[0.523, 0.863]0.679_[0.523,\,0.863] 0.708[0.546, 0.881]0.708_[0.546,\,0.881] 0.686[0.531, 0.863]0.686_[0.531,\,0.863] Occupation 1.962[1.690, 2.244]1.962_[1.690,\,2.244] 1.134[0.919, 1.408]1.134_[0.919,\,1.408] 0.956[0.786, 1.164]0.956_[0.786,\,1.164] 0.931[0.766, 1.128]0.931_[0.766,\,1.128] 0.990[0.822, 1.181]0.990_[0.822,\,1.181] Single Genre 4.360[3.538, 5.216]4.360_[3.538,\,5.216] 3.432[2.498, 4.527]3.432_[2.498,\,4.527] 2.649[1.984, 3.411]2.649_[1.984,\,3.411] 2.865[2.152, 3.655]2.865_[2.152,\,3.655] 2.677[1.961, 3.462]2.677_[1.961,\,3.462] Independent Users 2.431[2.079, 2.823]2.431_[2.079,\,2.823] 2.040[1.667, 2.464]2.040_[1.667,\,2.464] 1.899[1.554, 2.291]1.899_[1.554,\,2.291] 1.949[1.586, 2.355]1.949_[1.586,\,2.355] 1.904[1.568, 2.272]1.904_[1.568,\,2.272] Occupation + Genre + Decade 7.378[6.993, 7.779]7.378_[6.993,\,7.779] 8.138[7.701, 8.585]8.138_[7.701,\,8.585] 7.350[6.984, 7.738]7.350_[6.984,\,7.738] 7.383[7.003, 7.767]7.383_[7.003,\,7.767] 7.368[6.981, 7.745]7.368_[6.981,\,7.745] Models and baselines. We compare three analogical models: the Fisher-Rao (Proposition 12), Aitchison (Proposition 15) and the arithmetico-geometric analogy of 20. As baselines, we consider two leave-one-out predictors: the average distribution of the target category, and an additive model combining cohort and category marginal distributions. Metrics. Performance is evaluated using the Jensen Shannon divergence. Statistical significance is assessed through paired permutation tests over all hidden cohort-category pairs. Bootstrap confidence intervals are reported for all average performance measures. Conditions. We evaluate the proposed models under six experimental conditions of increasing difficulty. (1) Age: Users are partitioned according to their age group, and movie categories correspond to the standard MovieLens genres. (2) Age + Gender. Cohorts are refined by combining age group and gender. (3) Occupation. Cohorts are defined by the users’ occupation. Since MovieLens contains a larger number of occupations than age groups, this setting increases the diversity of user profiles and the complexity of the transfer task. (4) Single Genre. Users are partitioned into age groups. The evaluation is restricted to movies associated with a single genre, reducing taste ambiguity. (5) Independent Users. Users are randomly partitioned into disjoint training and evaluation subsets. This prevents the same users from contributing to both the prediction and the ground truth, thereby providing a stricter evaluation of the analogical transfer. (6) Occupation + Genre + Decade. The most challenging setting combines occupation-based cohorts with movie categories defined by genre and release decade, producing a much finer partition of the data and reducing the number of observations available for each distribution. Results. Table 1 reports the prediction performance on the six experimental settings. Across all experiments, the analogical approaches globally outperform both leave-one-out baselines. In particular, the gain over the genre-only baseline is substantial in the standard demographic transfer tasks, showing that the proposed analogies successfully transfer rating profiles between different user cohorts rather than relying solely on the target movie category. Among the proposed geometries, Fisher-Rao achieves the lowest average Jensen-Shannon divergence in five of the six experiments, while the Aitchison geometry performs best on the occupation-based setting. The arithmetico-geometric analogy remains consistently close to the two Riemannian approaches. The relatively small differences between the analogical models suggest that the proposed framework is robust with respect to the underlying geometry while consistently outperforming the non-analogical baselines. As expected, prediction becomes increasingly difficult as the transfer task is made more challenging. Moving from age-based cohorts to finer demographic partitions and more specific movie categories results in a gradual increase of the Jensen-Shannon divergence for all methods. Nevertheless, the analogical models preserve their advantage over the baselines in the first five experimental settings. The final experiment, combining occupation, genre and decade, represents the most challenging configuration. In this case, the performance of the analogical models becomes comparable to that of the genre-only baseline, suggesting that when both cohorts and categories become highly specific, the amount of transferable information naturally decreases. Overall, these experiments demonstrate that proportional analogies provide an effective mechanism for transferring preference distributions across demographic groups. The improvements are observed consistently across a broad range of experimental settings, indicating that the proposed analogical constructions capture meaningful structural relationships between categorical probability distributions. Conclusion We introduced an intrinsic proportional analogy relation on Riemannian manifolds based on the notion of geodesic midpoint. The proposed construction satisfies the fundamental properties of proportional analogies while naturally reflecting the geometry of the underlying manifold. We established several theoretical characterizations of this relation, including equivalent formulations on geodesically convex manifolds and invariance under isometries. We further showed that, on Riemannian symmetric spaces, the analogy admits a particularly simple form, leading to closed-form characterizations on several important manifolds such as spheres, hyperbolic spaces, the manifold of symmetric positive definite matrices, shape spaces, and manifolds of probability distributions. More generally, this work provides a unified geometric framework for extending proportional analogies beyond symbolic domains and Euclidean vector spaces. Several directions remain to be explored, including extensions to non-Riemannian geometries, the development of efficient numerical solvers on general manifolds, and empirical investigations of analogy-based learning methods. We believe that these developments could lead to new applications in transfer learning, meta-learning, sparse coding, and geometric data augmentation. 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