IntroductionThe comparison of heterogeneous data distributions is a fundamental task in computer vision, computational biology and machine learning. Most existing approaches rely on using a suitable ground cost function such as an available metric. A classic example is the Wasserstein distance which seeks an optimal transport (OT) between two given distributions. However, often the given distributions are in heterogeneous spaces, where a readily available ground cost function between these spaces does not generally exist. Additional effort may be required to learn appropriate cost functions (Cuturi & Avis, 2014;Heitz et al., 2021). However, even if there exists a natural embedding into a Proceedings of the 42 nd International Conference on Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s). . Joint (aligning) transfer of two metric spaces (gray surfaces with surface distance) to a fixed reference space, namely to the sphere and the torus by our method, where the color "yellow" corresponds to higher values, see Subsection 6.1.canonical joint metric space, this metric may not be suited to accurately gauge differences in their samples. Examples of such heterogeneous settings are, e.g., the comparison of graph-or mesh-valued data such as 3d shapes or manifolds. This paper introduces a novel framework based on OT which enables the joint comparison and visualization of heterogeneous datasets by optimally transferring them into an a-priori fixed metric space, see Figure 1 for an illustrative example.
Previous WorkAs a first step, we highlight the most relevant contributions related to our study among the vast literature on OT and dimensionality reduction.Optimal Transport with Invariances Classical Wasserstein distances allow to compare measures on a common metric space. In specific tasks in geometry processing like shape matching, the considered measures, however, live on distinct metric spaces that incorporate the different geodesic distances on the given shapes. To address this, Gromov-Wasserstein (GW) distances are introduced (Mémoli, 2011;Sturm, 2006). In (Alaya et al., 2022), an approximation of GW distances is obtained by jointly embedding measures into Euclidean spaces. Furthermore, projection-and subspace-robust Wasserstein-2 distances are introduced in (Paty & Cuturi, 2019). Another approach incorporates invariance to Euclidean isometries via the Wasserstein Procrustes problem (Grave et al., 2019). This is extended to account for linear operators with bounded Schatten norms in (Alvarez-Melis et al., 2019) and to Gaussian mixture applications in (Salmona et al., 2024). As outlined in Section 5, our paper extends such invariant OT to non-Euclidean domains.Joint Dimensionality Reduction Dimensionality reduction is a core topic in machine learning enabling visualization and clustering by finding optimal low-dimensional representations of data. Classical methods like principal component analysis (PCA) (Greenacre et al., 2022) and multidimensional scaling (MDS) (Carroll & Arabie, 1998) preserve large variations or pairwise distances, but fail on nonlinear manifolds (Alaya et al., 2022;Deng et al., 2024). Nonlinear approaches, e.g., locally linear embedding (Roweis & Saul, 2000), probabilistic models, e.g., t-distributed stochastic neighbor embedding (t-SNE) ( Van der Maaten & Hinton, 2008), and deep learning methods, e.g., variational autoencoders (VAEs), (Kingma et al., 2019) address these challenges. As an extension, several recent methods focus on joint embeddings of heterogeneous data. Here, we are given data on two incompatible domains and are interested in simultaneous embedding. The manifold-aligning generative adversarial network (Amodio & Krishnaswamy, 2018) employs a generative adversarial network for domain alignment. Maximum mean discrepancy (MMD) manifoldalignment (Liu et al., 2019) balances an MMD and a distortion term. UnionCom (Cao et al., 2020) leverages the generalized unsupervised manifold alignment (GUMA) (Cui et al., 2014). Single-cell alignment with optimal transport (SCOT) (Demetci et al., 2022) and the partial manifold alignment algorithm (Cao et al., 2022) employ GW distances. Finally, the recently proposed joint multidimensional scaling (JMDS) (Chen et al., 2023) algorithm combines MDS with OT for the joint embedding of two datasets into a shared Euclidean space. Complementing this and research on non-Euclidean embeddings (McInnes et al., 2018;Deng et al., 2024) and connections between MDS and GW (Van Assel et al., 2024;Clark et al., 2025), we propose a joint embedding into arbitrary metric spaces based on factored transport plans (Forrow et al., 2019) and GW distances.
ContributionOur main contributions are the following: 1. Towards the aligned "embedding" of heterogeneous metric spaces into a fixed, not necessarily Euclidean space, we propose to minimize an unbalanced OT problem with a quadratic cost function, where the marginals are penalized by GW distances. This formulation seeks near-isometric joint embeddings of the inputs into a metric space, while enabling an optimal comparison in the Wasserstein distance.2. We prove that our functional has a minimizer. Further, if its regularization parameter goes to infinity, our functional approaches the so-called "embedded Wasserstein distance" (Salmona et al., 2024). In this sense, we also refer to our model as the "relaxed embedded Wasserstein distance". 3. For the (approximate) computation of a minimizer, we provide an equivalent formulation of our model as a quadratic, multi-marginal, unbalanced OT problem. A biconvex relaxation enables the application of existing algorithms to solve the problem numerically. 4. We recall JMDS from the point of view of our model: while we are searching for the weight of atomic measures fixing their supports, JMDS fixes the weights and aims to find the supports of the measures. 5. Numerical experiments demonstrate the potential of our method for joint embeddings of heterogeneous data on the 2d Euclidean space, the torus, the 2-sphere and the space of 2d Gaussians with the Wasserstein distance.All proofs are given in Appendix A.
OT-Based DistancesGiven a compact metric space (Z, d Z ), we denote by P(Z) the space of probability measures defined on the Borel-σalgebra induced by the distance d Z . For two compact metric spaces Z 1 and Z 2 , the push-forward measure of µ ∈ P(Z 1 ) under a measurable map T : Z 1 → Z 2 is denoted byT ♯ µ := µ • T -1 ∈ P(Z 2 ).The set of transport plans between two measures µ 1 ∈ P(Z 1 ) and µ 2 ∈ P(Z 2 ) is given byΠ(µ 1 , µ 2 ) := π ∈ P(Z 1 × Z 2 ) : P i,♯ π = µ i , i = 1, 2 ,whereP i : Z 1 × Z 2 → Z i , (z 1 , z 2 ) → z i , i = 1, 2, denotes the projection to the first and second component, respectively.To lift d Z from Z to P(Z), we rely on the Wasserstein(-2) distance between µ 1 , µ 2 ∈ P(Z) defined byW(µ 1 , µ 2 ) := min π∈Π(µ1,µ2) Z×Z d 2 Z (z, z ′ ) dπ(z, z ′ ) 1 2 . (1)By Π o (µ 1 , µ 2 ), we denote the set of minimizers in (1). The Wasserstein distance metricizes the weak convergence of measures, where a sequence of measures (µ n ) n∈N converges weakly to a measure µ ∈ P(Z), written µ n ⇀ µ, if for every (bounded) continuous function φ : Z → R, we have Z φ dµ n → Z φ dµ as n → ∞.To compare heterogeneous data via their internal geometry, we can rely on GW distances, which enables us to compare measures on different metric spaces. To highlight the connection between measures and underlying spaces, we consider metric measure spaces (mm-spaces), which are triples X = (X, d X , ξ) such that (X, d X ) is a compact metric space and ξ ∈ P(X). Two mm-spaces X i = (X i , d Xi , ξ i ), i = 1, 2, are isomorphic if there exists a bijective map I : supp ξ 1 → supp ξ 2 between the supports of the measures such that I is measure-preserving, i.e., I ♯ ξ 1 = ξ 2 , andI is an isometry, i.e., d X1 (x 1 , x ′ 1 ) = d X2 (I(x 1 ), I(x ′ 1 )) for all x 1 , x ′ 1 ∈ supp ξ 1 . By [X]we denote the equivalence class of all mm-spaces which are isomorphic to X. Finally, if I is an isometry between two metric spaces X i , i = 1, 2, we write I : X 1 → X 2 , and just X 1 → X 2 if such an isometry between X 1 and X 2 exists.Sturm's GW distance (Sturm, 2006) between two mmspaces X i , i = 1, 2, is defined bySGW(X 1 , X 2 ) := inf (Z,d Z ) metric space, Ii : supp ξi →Z, i=1,2 W(I 1,♯ ξ 1 , I 2,♯ ξ 2 ),where the Wasserstein distance is taken over the respective space (Z, d Z ). The SGW distance seeks optimal isometric embeddings of either mm-space into a joint metric space such that their Wasserstein distance in the embedded space is minimal, see Figure 2a. Indeed, SGW is a metric on the equivalence classes of mm-spaces.Mémoli proposed a different GW distance, which is numerically more appealing than Sturm's construction. For two mm-spaces X i , i = 1, 2, Mémoli's GW distance (Mémoli, 2011) is defined byGW(X 1 , X 2 ) := min γ∈Π(ξ1,ξ2) G X1,X2 (γ) 1 2with the quadratic GW objectiveG X1,X2 (γ) := (X1×X2) 2 d X1 (x 1 , x ′ 1 ) -d X2 (x 2 , x ′ 2 ) 2 × dγ(x 1 , x 2 ) dγ(x ′ 1 , x ′ 2 ).The problem seeks a transport between X 1 and X 2 which minimizes the overall pairwise distance distortion, see Figure 2b. Both distances-SGW and GW-define a metric on the equivalence classes of mm-spaces. Furthermore, both distances induce the same topology, but SGW turns these equivalence classes into a complete metric space, while GW does not (Sturm, 2006;Mémoli, 2011). The later can be alleviated by embedding into a larger space and extending the GW definition accordingly (Sturm, 2023).
Unbalanced OT With GW PenalizationThroughout this section, let (Z, d Z ) be a fixed compact metric space. Furthermore, let X i = (X i , d Xi , ξ i ), i = 1, 2, be two mm-spaces, whose measures' support can be isometrically embedded into (Z, d Z ), i.e., supp ξ i → Z.For this specific setting, we relax Sturm's GW distance to EW(X 1 , X 2 ) := infI1 : supp ξ1 →Z I2 : supp ξ2 →Z W(I 1,♯ ξ 1 , I 2,♯ ξ 2 ). (2)On a certain subspace of equivalence classes of mm-spaces, this defines a metric, which we call embedded Wasserstein metric. For the specific case of Euclidean spaces X 1 , X 2 , and Z, this reduces to the Wasserstein Procrustes problem (Grave et al., 2019) and to the "embedded Wasserstein metric" in (Salmona et al., 2024). We adopt this name.Proposition 3.1. EW defines a metric on the subset of isomorphic classes [(X, d X , ξ)] for which there exist surjective isomorphism I : supp ξ → Z.By the following proposition, the infimum in (2) is attained.Proposition 3.2. Let X i , i = 1, 2 be two mm-spaces and (Z, d Z ) a metric space such that supp ξ i → Z for i = 1, 2.Then the infimum in (2) is attained.While EW relies on appropriate isometries, the following relaxation enables us to handle arbitrary mm-spaces X i , i = 1, 2. For λ > 0, we consider the following GW penalized unbalanced OT problemEW λ (X 1 , X 2 ) := inf π∈P(Z×Z) Z×Z d 2 Z (z, z ′ ) dπ(z, z ′ ) + λ 2 i=1 GW 2 (X i , (Z, d Z , P i,♯ π)) 1 2 . (3)By penalizing the GW terms, the marginals P i,♯ π ∈ P(Z) take a form that enforces (Z, d Z , P i,♯ π) to be nearly isomorphic to the inputs X i , i = 1, 2. Furthermore, the first term ensures that the marginals P i,♯ π, i = 1, 2 are close in the Wasserstein distance on (Z, d Z ). The penalization extends EW to non-isomorphic metric spaces by considering minimum-distortion embeddings. By the following proposition, the infimum in (3) is attained.Proposition 3.3. Let X i , i = 1, 2 be two mm-spaces and (Z, d Z ) a metric space. Then (3) admits a solution.As the GW penalization enforces isometry, EW in (2) becomes the limit of EW λ in (3) if λ goes to infinity.Proposition 3.4. Let X i , i = 1, 2 be two mm-spaces and let (Z, d Z ) be a metric space such that supp ξ i → Z. Let (λ n ) n∈N be a sequence with λ n → ∞ as n → ∞. Then any sequence (π n ) n∈N of minimizers of EW λn (X 1 , X 2 ) converges weakly, up to a subsequence, to some π ∈ P(Z × Z).There exist isometries(I 1 , I 2 ) realizing EW(X 1 , X 2 ) such that π ∈ Π o (I 1,♯ ξ 1 , I 2,♯ ξ 2 ).If there exist no isometric embeddings supp ξ i → Z, the limit of EW λ as λ → ∞ yields best possible approximations of X i on the given metric space (Z, d Z ) with respect to the GW distance. More precisely, a GW approximation of an mm-space X on the metric space(Z, d Z ) is a minimizer ζ ∈ P(Z) of GWA(X) := min ζ∈P(Z) GW 2 (X, (Z, d Z , ζ)).(4)For discrete mm-spaces, GW approximations are studied in (Clark et al., 2025) and are closely related to MDS.Proposition 3.5. Let X i , i = 1, 2, be two mm-spaces and let (Z, d Z ) be a metric space. Let (λ n ) n∈N be a sequence with λ n → ∞ as n → ∞. Then any sequence (π n ) n∈N of minimizers of EW λn (X 1 , X 2 ) converges weakly, up to a subsequence, to π ∈ Π(ζ 1 , ζ 2 ), where ζ i ∈ P(Z) is a GW approximation of X i .On the other side, if λ → 0, the influence of the Wasserstein term in EW λ increases. Figuratively, the marginals P 1,♯ π and P 2,♯ π of the minimizers π of EW λ have to become more similar if λ goes to zero. In the limit case, when both marginals coincide, we obtain a fixed-support GW barycenter (Beier et al., 2023). In detail, a fixed-support GW barycenter on the metric space(Z, d Z ) is a minimizer ζ ∈ P(Z) of GWB(X 1 , X 2 ) := min ζ∈P(Z) 2 i=1 GW 2 (X i , (Z, d Z , ζ)). (5)Proposition 3.6. Let X i , i = 1, 2, be two mm-spaces and let (Z, d Z ) be a metric space. Let (λ n ) n∈N be a sequence with λ n → 0 as n → ∞. Then any sequence (π n ) n∈N of minimizers of EW λn (X 1 , X 2 ) converges weakly, up to a subsequence, to (Id, Id) ♯ ζ, where ζ ∈ P(Z) is a fixedsupport GW barycenter.In the rest of the paper, we skip the integration domains of the integrals for better readability, since they are clear from the context. Then, by definition of the GW distance, EW λ in (3) can be rewritten asEW 2 λ (X 1 , X 2 ) = inf π∈P(Z×Z) inf γi∈Π(ξi,P Z,♯ π) i=1,2 d 2 Z (z, z ′ ) dπ(z, z ′ ) + λ G X1,Z (γ 1 ) + G X2,Z (γ 2 ) (6)or equivalently asEW 2 λ (X 1 , X 2 ) = inf µ1,µ2∈P(Z) inf π∈Π(µ1,µ2) inf γi∈Π(ξi,µi) i=1,2 d 2 Z (z, z ′ ) dπ(z, z ′ ) + λ G X1,Z (γ 1 ) + G X2,Z (γ 2 ) .(7)For computing EW λ , we will rewrite the term by 4-plans.To this end, we use the notation Z 1 = Z 2 := Z and denote the projection onto X i ,Z i by P Xi , P Zi , respectively. Now we consider 4-plansα ∈ P(X 1 , Z 1 , Z 2 , X 2 ) fulfilling P Xi,♯ α = ξ i , i = 1, 2. Let π := P Z1×Z2,♯ α and γ i := P Xi×Zi,♯ α, i = 1, 2. (8)Clearly, such plans automatically fulfillP Zi,♯ π = P Zi,♯ γ i , i = 1, 2see Figure 3. Thus, (6) can be reformulated as a quadratic, multi-marginal, unbalanced OT problemEW λ (X 1 , X 2 ) = inf α∈P(X1×Z1×Z2×X2) P X i ,♯ α=ξi,i=1,2 F λ (α) 1 2 (9)with the quadratic objectiveF λ (α) := 1 2 d 2 Z (z 1 , z 2 ) + d 2 Z (z ′ 1 , z ′ 2 ) + λ 2 i=1 (d Xi (x i , x ′ i ) -d Z (z i , z ′ i )) 2 × dα(x 1 , z 1 , z 2 , x 2 ) dα(x ′ 1 , z ′ 1 , z ′ 2 , x ′ 2 ).We summarize our findings in the following proposition.Proposition 3.7. If α solves (9), then its projections (8) are solutions of (6) and in particular π is a solution of (3).Conversely, any solution of (3) can be expressed this way.In Appendix B, we illustrate relations between the Wasserstein distance, GW, EW and EW λ by numerical examples.
Bi-Convex RelaxationAt its core, the computation of EW λ in (9) requires the solution of a quadratic optimization problem. Similar formulations appear, for instance, in the computation of the GW distance (Peyré et al., 2016;Séjourné et al., 2021), in the multi-marginal GW setting (Beier et al., 2023), and in CO-OT (Vayer et al., 2020b). All of these quadratic OT problem have in common that they can be numerically solved using a block-coordinate descent on their bi-convex relaxations. In the following, we adapt this approach to our multi-marginal, unbalanced transport problem (9).For this, we decouple the minimization with respect to the inner and outer transport plan γ in the double integral of (9). More precisely, denoting the inner plan by α 1 and the outer plan by α 2 , we consider the bi-convex relaxationinf α1,α2∈P(X1×Z1×Z2×X2) (P X i ) ♯ α k =ξi, i,k=1,2 F λ (α 1 , α 2 )(10)with the bilinear objectiveF λ (α 1 , α 2 ) := 1 2 d 2 Z (z 1 , z 2 ) + d 2 Z (z ′ 1 , z ′ 2 ) + λ 2 i=1 (d Xi (x i , x ′ i ) -d Z (z i , z ′ i )) 2 × dα 1 (x 1 , z 1 , z 2 , x 2 ) dα 2 (x ′ 1 , z ′ 1 , z ′ 2 , x ′ 2 ).By construction, the minimizers of (10) constitute a lower bound to the original, quadratic problem (9). Moreover, every bi-convex minimizer of the form α 1 = α 2 yields a minimizer of the original problem.In order to find a numerical solution, we apply blockcoordinate descent, which consists of alternatively fixing α 1 and α 2 in (10) and minimizing with respect to the other argument. Fixing α 1 , it remains to solve the (linear) multimarginal, unbalanced OT problem infP X i ,♯ α2=ξi i=1,2 c 1 (x ′ 1 , z ′ 1 , z ′ 2 , x ′ 2 ) dα 2 (x ′ 1 , z ′ 1 , z ′ 2 , x ′ 2 ),(11)where the effective cost is given byc 1 (x ′ 1 , z ′ 1 , z ′ 2 , x ′ 2 ) := 1 2 d 2 Z (z ′ 1 , z ′ 2 ) + λ 2 i=1 d Xi (x i , x ′ i ) -d Z (z i , z ′ i ) 2 dα 1 (x 1 , z 1 , z 2 , x 2 ). (12)The cost function additively decouples into three partial cost functions solely depending on two coordinates of c 1 . Unbalanced, multi-marginal OT formulations of this kind are treated in (Beier et al., 2022), where, relying on an entropic regularization, a multi-marginal Sinkhorn scheme is proposed. Mathematically, this means that, for a regularization parameter ε > 0, we approximate the minimizer of (11) by infP X i ,♯ α2=ξi i=1,2 c 1 dα 2 + ε KL(α 2 , υ),where υ denotes the uniform measure on X 1 ×Z 1 ×Z 2 ×X 2 and KL the Kullback-Leibler divergence. More precisely, with the Radon-Nikodým derivative dγ/ dυ, the KL divergence is given by KL(γ, υ) := log( dγ/ dυ) dυ if γ ≪ υ and KL(γ, υ) := ∞ otherwise. In total, we may approximate the minimizer of the quadratic, unbalanced, multimarginal OT problem (9) by applying Algorithm 1.Algorithm 1 Computation of EW λ input X i = (X i , d Xi , ξ i ), i = 1, 2. (mm-spaces) input (Z, d Z ) (finite metric space) input λ > 0 (penalization parameter) input ε > 0 (regularization parameter) 1: Initialize α 1 = α 2 := ξ 1 ⊗ µ ⊗ µ ⊗ ξ 2where µ is the uniform measure on (Z, d Z ) 2: while not converged do 3:Compute c 1 as in (12) 4: Update α 2 using the multi-marginal Sinkhorn scheme in (Beier et al., 2022) with c 1 5:Compute c 2 analogous to (12) 6:Update α 1 using the multi-marginal Sinkhorn scheme in (Beier et al., 2022) with c 2 7: end while output α 1 or α 2
Discrete EW λ and JMDSThe distance EW in (2) generalizes Wasserstein Procrustes (Grave et al., 2019) to non-Euclidean spaces, while EW λ in (3) is closely related to the JMDS model (Chen et al., 2023). In this section, we explain the relation.We consider the discrete case, whereX 1 := {x 1 1 , . . . , x n11 } and X 2 := {x 1 2 , . . . , x n2 2 } are point sets equipped with dissimilarity distances d X1 and d X2 as well as measuresξ 1 := n1 j=1 ξ j 1 δ x j 1 , ξ 2 := n2 j=1 ξ j 2 δ x j 2 .Discrete EW λ fixes a discrete embedding spaceZ 1 = Z 2 = Z := {z 1 , . . . , z m }.with metric d Z . Accordingly, we use measures with fixed supportsµ i = m j=1 µ j i δ z j , γ i = ni,m j,k=1 γ j,k i δ x j i ,z k , π = m j,k=1 π j,k δ z j ,z k ,where i = 1, 2. Since the supports of the measures are fixed, we can restrict ourselves to the weight matricesµ i ∈ ∆ m , ξ i ∈ ∆ ni , π ∈ Π(µ 1 , µ 2 ) and γ i ∈ Π(ξ i , µ i ), i = 1, 2 in the corresponding probability simplicies ∆. Then EW λ in (7) becomes the discrete minimization problemEW λ (X 1 , X 2 ) = min µ i ∈∆m i=1,2 min π∈Π(µ1,µ2) min γi∈Π(ξi,µi) i=1,2 π, D 2 Z,Z + λ G X1,Z (γ 1 ) + G X2,Z (γ 2 ) ,whereD 2 Z,Z := d 2 Z (z j , z k ) m j,k=1 and, for i = 1, 2, G Xi,Z (γ i ) := ni,m j,k=1 ni,m r,s=1 γ s,k i γ j,k i d X1 (x j i , x r i ) -d Z (z k , z s ) 2 .JMDS aims to find point setsZ 1 := {z 1 1 , . . . , z n1 1 } ⊂ R d , Z 2 := {z 1 2 , . . . , z n2 2 } ⊂ R dsuch that the dissimilarity relations in X i are approximately preserved in Z i , i = 1, 2, while ensuring that the intermediate points are optimally aligned. Here, Z i , i = 1, 2 are exclusively in R d with the Euclidean metric d(x, y) = ∥x-y∥.Instead of measures with fixed support, we useµ i := ni j=1 1 n i δ z j i , π = m j,k=1 π j,k δ z j 1 ,z k 1 ,as well asγ i := ni j,k=1 γ j,k i δ x j i ,z k i with γ j,k i := 1 ni j = k, 0 j ̸ = k,and will optimize over the supports Z i , i = 1, 2. Then (7) becomesJMDS λ (X 1 , X 2 ) := min Z1,Z2 min π∈Π( 1 n 1 1n 1 , 1 n 2 1n 2 ) π, D 2 Z1,Z2 + λ G 1 (Z 1 ) + G 2 (Z 2 ) , (13)whereD 2 Z1,Z2 := d 2 (z j 1 , z k 2 )n1,n2 j,k=1 and, for i = 1, 2,G i (Z i ) := ni j,k=1 ni r,s=1 γ j,k i γ r,s i d Xi (x j i , x r i ) -d(z k i , z s i ) 2 = ni j,k=1 1 n 2 i d Xi (x j i , x k i ) -d(z j i , z k i ) 2 .This is exactly the functional proposed as JMDS in (Chen et al., 2023). Note that the Wasserstein Procrustes model is given bymin π∈Π( 1 n 1 1n 1 , 1 n 2 1n 2 ) min Q∈SO(d) π, D 2 QZ1,Z2 .This is exactly the first summand in JMDS λ , where the minimization over the special orthogonal group SO(d) can be skipped in ( 13) since G i (Z i ), i = 1, 2, are invariant under orthogonal transforms.To summarize: 1. Our discrete EW λ fixes the support of the marginals µ i in (7) which results in the optimization over the weights, where JMSD fixes the weights of the µ i and optimizes over the supports. 2. Fixing the support has the advantage that we can work with arbitrary metric spaces (Z, d Z ), while the "free support" approach of JMDS is restricted to the Euclidean space. 3. The minimization problems have to be tackled by completely different optimization algorithms, namely an unbalanced multimarginal Sinkhorn algorithm for the block-coordinate descent in Algorithm 1 for EW λ and the so-called SMACOF method combined with Wasserstein Procrustes minimizations (Chen et al., 2023) for JMDS.
Numerical ResultsNext, we provide several proof-of-concept examples1 . Additional parameter ablations are described in Supplement C.
Joint Embedding of 3d ShapesIn the first example, we exploit EW λ to align and embed 3d shapes-the surfaces of objects in R 3 -into a joint space (Z, d Z ). For this, we interpret a 3d shape as mm-space X := (X, d X , ξ), where X is the surface, d X is the surface (or geodesic) distance, and ξ is the uniform measure. Practically, X is parametrized by the vertices of a triangular mesh, d X is approximated using Dijkstra's algorithm (Dijkstra, 1959) on the corresponding graph, and ξ is chosen as the discrete uniform measure. For the joint embedding of two given (discrete) shapes X i , i = 1, 2, we compute the 4-plan α in (9) using the discretization in Section 5 and Algorithm 1. Since the relaxation behind EW λ does not yield an isometry, but only a transport plan γ i = P Xi×Zi,♯ α, we visualize the computed relaxed embeddings by the marginals P Zi,♯ γ i = P Zi,♯ α.Bended Rectangles We start by embedding an S-bended rectangle and a Swiss roll (with and without a hole) into R 2 . Intuitively, we expect that the surfaces without holes are unrolled by the relaxed embedding behind EW λ , since there  Figure 5. Embedding and alignment of human shapes from the FAUST dataset into R 2 . For our method, we visualize the marginals P Z i ,♯ α of the computed 4-plan α in (9) and compare them with JMDS. Here JMDS tends to split some of the extremities.actually exists isometric embeddings. For the discretization, we choose Z as an equispaced 50×50 grid on [0, 1.3] 2 ⊂ R 2 and d Z as the corresponding Euclidean distance and apply Algorithm 1 with λ = 100 and ε = 10 -3 . In Figure 4, the relaxed embeddings are visualized by P Zi,♯ α , where the point sizes represent the underlying probability masses/weights. As comparison, we also show the results of JMDS with λ = 10 and ε = 10 -3 for the incorporated regularized OT problem. Both methods produce representations that unroll the shapes and succeed in aligning the embeddings. JMDS, however, produces an unexpected hole during the alignment of the S-bended shape and the Swiss role with hole.Human Shapes Since we are not limited to isometries, we next consider an experiment where isometric embeddings Ours (λ = 1) 0.0057±0.004 0.017±0.007 JMDS (λ = 1) 0.0071±0.007 0.029±0.010Ours (λ = 10) 0.0048±0.004 0.027±0.016 JMDS (λ = 10) 0.0057±0.005 0.046±0.023Table 1. Mean and standard deviation of GW distance (between input shapes and embeddings) and of Wasserstein distance (between both embeddings) for joint 2d embeddings of the FAUST dataset via our method and JMDS. downsampled to 35 vertices, and both methods are regularized with ε = 0.001. For our method, we set Z according to a uniform 30×30 grid on [0, 1.3]2 . Results are shown in Table 1. In total, we here achieve better joint embeddings than JMDS both in terms of GW and Wasserstein.Spherical and Toroidal Embeddings Finally, we consider the alignment of spherical and toroidal subsets, i.e., the joint embedding into a non-Euclidean space. More precisely, we consider the 3d shapes in the introductory Figure 1. As target space (Z, d Z ), we choose a 30×30 grid on the canonical parametrization of the sphere S(2) and the torus T(2) = S(1) × S(1) equipped with the corresponding geodesic distance. Using λ = 10 3 and ε = 10 -3 for Algorithm 1, we embed the spherical rectangle and a spherical cap, both with a hole, onto the sphere, and the half-torus and toroidal triangle onto the torus. The relaxed embeddings are shown in Figure 1, where the color encodes the mass of the computed marginals. Up to a smoothing due to the incorporated entropic regularization, the transport plans γ i = P Xi×Zi,♯ α correspond to the expected isometries.
Alignment of Feature SpacesIn the next example, we use our method to align different feature spaces occurring in real-world data. More precisely, we consider the genetyping-the determination of the corresponding cell class-of single cells from various measured modalities. Multimodal technologies as proposed in (Cheow et al., 2016;Chen et al., 2019) are, however, uncommon such that there is a rising interest in embedding the corresponding features into a joint feature space (Demetci et al., 2022;Liu et al., 2019;Chen et al., 2023). Following the experiments of (Demetci et al., 2022) for n single cells, our first aim is to assign the recorded features X 1 := {x 1 1 , . . . , x n 1 } ⊂ R d1 from one modality to the features X 2 := {x 1 2 , . . . , x n 2 } ⊂ R d2 of another modality, i.e., to recover the underlying one-toone correspondence between x j 1 and x j 2 . Our second goal consists in transferring a classifier from X 1 to X 2 in an unsupervised manner.To achieve both goals, we apply the following methodology: 1. We equip the recorded, uncorrelated features X 1 ⊂ R d1 and X 2 ⊂ R d2 with specific distances d Xi estimated with the unsupervised SCOT routine (Demetci et al., 2022). 2. We equip the constructed metric spaces with the uniform measure to obtain the mm-spaces X i := (X i , d Xi , ξ i ).3. Fixing a 20×20 grid Z := {z 1 , . . . , z m } ⊂ R 2 over [0, 1] 2 equipped with the Euclidean metric, we apply Algorithm 1 with the discretization in Section 5 to compute the 4-plan α in ( 9). 4. Based on the transport plans γ i := P Xi×Zi,♯ α, we pair x j i with its barycentric projection xj i ∈ R m given byxj i := m k=1 γ jk i z k m k=1 γ jk i .For the first aim, the identification of the underlying correspondence, we identify x j 1 with x k 2 whose barycentric projection xk 2 is closest to xj 1 . To quantify the quality of the pairing, we rely on the FOSCTTM (fraction of samples closer than the true match) score (Liu et al., 2019). More precisely, FOSCTTM is defined by1 n 2 n j=1 # k ∈ {1, . . . , n} : ∥x j 1 -xk 2 ∥ < ∥x j 1 -xj 2 ∥and ranges from 0 (perfect identification) to 1.For the second aim, the transfer of a classifier from X 1 to X 2 , we exemplarily consider the k-nearest neighbor (KNN) method. More precisely, we use the barycentric projections {x 1 1 , . . . , xn 1 } and the corresponding gene type labels of the first modality to classify a feature x k 2 (more exact xk 2 ) from the second modality, where we consider the closest five neighbors. Here, higher classification accuracies (KNN-Acc) indicate better class alignments.For the experiments, we employ the publicly available datasets 2 from (Demetci et al., 2022), where SNAREseq   The computed embedding of X i into R 2 of our method and JMDS are visualized in Figure 6. For FOSCTTM and KKN-Acc, we additionally compare both methods with the alignment techniques UnionCom (Cao et al., 2020) and SCOT (Demetci et al., 2022) using the same distances on X i . Note that UnionCom relies on GUMA and t-SNE, whereas SCOT relies on GW and MDS. The results are recorded in Table 2. The hyperparameters of all methods are chosen according to a grid search minimizing the FOSCTTM on a 10% validation split, see Supplement D. A further experiment with different feature spaces for MNIST and FashionMNIST is given in Supplement E.
Alignment of Gaussian Mixture ModelsIn the final example, inspired by (Salmona et al., 2024), we want to align Gaussian mixture models (GMMs). For this, we consider the space of 2d Gaussians N 2 = {N (µ, Σ)|µ ∈ R 2 , Σ ∈ R 2×2 spd.} equipped with the Wasserstein distance, which yields a non-Euclidean geometry. A GMM can now be interpreted as mm-space X = (N 2 , W, ξ) with discrete measure ξ := n1 j=1 ξ j δ µj ,Σj . For the experiment, we fit two GMMs to affine-transformed 2d datasets using the expectation-maximization algorithm. Here, we employ the datasets "Blobs" and "Moons" from scikit-learn (Pedregosa et al., 2011), see Figure 7. For the alignment of the GMMs, we choose Z ⊂ N 2 , where the means form a 15×15 grid over [0, 1] 2 and we use a coarse grid over suitable matrices. Particularly, we consider matricesΣ = r 2 σ 2 1 σ 12 σ 12 σ 2 2 ,where we choose σ 2 1 , σ 2 2 ∈ {0.8, 1.} and σ 12 ∈ {-0.2, 0., 0.2}. The parameter r 2 > 0 corresponds to the mean variance of the considered GMMs. Applying Algorithm 1 to compute α in (9), and considering the marginals ζ i := P Zi,♯ α, which are again discrete measures on N 2 , we obtain the aligned GMMs in Figure 7.
ConclusionWe propose an unbalanced OT framework with GW marginal penalization which enables the joint embedding of two datasets based on pairwise intra-dataset distances. While our model can handle the aligned transfer into arbitrary metric spaces, it relies on their appropriate definition.In particular, due to computational restrictions, working on a grid in R d is restricted to small dimensions d. As a future research direction, we are interested in developing a non-Euclidean free-support solver based on existing free-support GW barycenter algorithms (Peyré et al., 2016;Vayer et al., 2020a;Beier & Beinert, 2025). This may combine our EW λ approach with a non-Euclidean version of JMDS.From a theoretical point of view, the joint embedding using proposed variational formulation may be adapted to handle multiple input spaces. The main issue is hereby the selection of an appropriate generalization of the employed Wasserstein part. A possible choice would be to utilize only the Wasserstein distance between the embeddings of the kth and (k + 1)st embedding. Since the employed multi-marginal Sinkhorn method linearly scales in the number of spaces, we expect that the resulting method remains numerically tractable for small numbers of spaces. The actual choice of the model and the numerical implementation of a multiple space variant, however, requires further studies.Let (π n ) n∈N ⊂ P(Z × Z) and π ∈ P(Z × Z) be such that π n ⇀ π as n → ∞. We handle the terms of F separately. Since d 2 Z is continuous in both arguments, we obtain by definition of the weak convergence thatZ×Z d 2 Z (z, z ′ ) dπ n (z, z ′ ) → Z×Z d 2 Z (z, z ′ ) dπ(z, z ′ )as n → ∞. We turn to the GW terms. For i = 1, 2, set µ i,n := P i,♯ π n and µ i := P i,♯ π.Then the weak lower semi-continuity of the marginal projection operators yieldsµ i,n ⇀ µ i as n → ∞. Now let γ i,n ∈ Π(ξ i , µ i,n ) be a solution of GW(X i , (Z, d Z , µ i,n )), n ∈ N. As (γ i,n ) n∈N ⊂ P(X i × Z) and the latter is weakly compact, we can choose a converging subsequence again denoted by (γi, n) n∈N , such that γ i,n ⇀ γ i ∈ P(X i × Z). Using again the weak continuity of the marginal projection, we get γ ∈ Π(ξ i , µ i ). Therefore, it holdsGW 2 (X i , (Z, d Z , P i,♯ π)) = GW 2 (X i , (Z, d Z , µ i )) ≤ (Xi×Z) 2 (d Xi (x i , x ′ i ) -d Z (z, z ′ )) 2 dγ(x i , z) dγ(x ′ i , z ′ )and by continuity of the integrand and the fact that(γ i,n ⊗ γ i,n ) ⇀ (γ i ⊗ γ i ), see, e.g., (Bogachev, 2018) (Prop. 2.7.8) furtherGW 2 (X i , (Z, d Z , P i,♯ π)) ≤ lim n→∞ (Xi×Z) 2 (d Xi (x i , x ′ i ) -d Z (z, z ′ )) 2 dγ i,n (x i , z) dγ i,n (x ′ i , z ′ ) = lim n→∞ GW 2 (X i , (Z, d Z , µ i,n )) = lim n→∞ GW 2 (X i , (Z, d Z , P i,♯ π n )).This gives lower semi-continuity of the functional of F . □ Proof of Proposition 3.4 1. First, we show thatEW λ (X 1 , X 2 ) ≤ EW(X 1 , X 2 ), λ > 0.(14)To this end, let I i : X i → Z i be isometries realizing EW(X 1 , X 2 ), i.e., EW(X 1 , X 2 ) = W(I 1,♯ ξ 1 , I 2,♯ ξ 2 ).By definition of the GW distance, we have GW(X i , (Z i , d Zi , I i,♯ ξ i ) = 0, i = 1, 2. Then we obtain for π ∈ Π 0 (I 1,♯ ξ 1 , I 2,♯ ξ 2 ) thatEW λ (X 1 , X 2 ) ≤ Z1×Z2 d 2 Z (z, z ′ ) dπ(z, z ′ ) 1 2 = W(I 1,♯ ξ 1 , I 2,♯ ξ 2 ) = EW (Z,d Z ) (X 1 , X 2 ).2. Now let (λ n ) n∈N with λ n → ∞ as n → ∞, and let π n ∈ P(Z × Z) realize EW λn (X 1 , X 2 ). Then we conclude by ( 14) thatEW(X 1 , X 2 ) ≥ EW λn (X 1 , X 2 ) = Z×Z d Z (z, z ′ ) dπ n (z, z ′ ) + λ n 2 i=1 GW(X i , (Z, d Z , P i,♯ π n )) ≥ λ n 2 i=1 GW(X i , (Z, d Z , (P i ) # π n )).Hence, we obtain for i = 1, 2 thatGW(X i , (Z, d Z , (P i ) # π n )) → 0 as n → ∞.Since (π n ) n∈N is contained in the weakly compact set P(Z × Z), there exists a subsequence (denoted in the same way) which weakly converges to some π ∈ P(Z × Z). Then also P i,♯ π n converges weakly to P i,♯ π, i = 1, 2 and we obtain as in the proof of (3.3), that GW(X i , (Z,d Z , P i,♯ π)) ≤ lim n→∞ GW(X i , (Z, d Z , P i,♯ π n )) = 0.Thus, there exist isometries I i : X i → Z with P i,♯ π = I i,♯ ξ i , i = 1, 2, i.e., π ∈ Π(I 1,♯ ξ 1 , I 2,♯ ξ 2 ). Finally, we obtain for any fixed λ > 0 thatEW(X 1 , X 2 ) ≤ Z1×Z2 d 2 Z (z 1 , z 2 ) dπ(z 1 , z 2 ) 1 2 = Z1×Z2 d 2 Z (z 1 , z 2 ) dπ(z 1 , z 2 ) + λ 2 i=1 GW(X i , (Z, d Z , P i,♯ π)) =0 1 2 ≤ lim n→∞ Z1×Z2 d 2 Z (z 1 , z 2 ) dπ n (z 1 , z 2 ) + λ 2 i=1 GW(X i , (Z, d Z , P i,♯ π n )) 1 2 ≤ lim n→∞ EW λn (X 1 , X 2 ) ≤ EW(X 1 , X 2 ). Hence π ∈ Π(I 1,♯ ξ 1 , I 2,♯ ξ 2 ) and EW(X 1 , X 2 ) = inf Ji:Xi →Z,i=1,2 W(J 1,♯ ξ 1 , J 2,♯ ξ 2 ) = Z1×Z2 d 2 Z (z 1 , z 2 ) dπ(z 1 , z 2 ) 1 2implies that (I 1 , I 2 ) minimizes EW(X 1 , X 2 ) and π realizes W(I 1,♯ ξ 1 , I 2,♯ ξ 2 ). □Proof of Proposition 3.5 Without loss of generality, assume that π n converges to π ∈ Π(ζ 1 , ζ 2 ) with ζ i ∈ P(Z) weakly; otherwise take a convergent subsequence. Since (Z,d Z ) is compact, the diameter M := sup z,z ′ ∈Z d(z, z ′ ) is finite. Using plans πn ∈ Π(ζ 1,n , ζ 2,n ), where ζ i,n ∈ P(Z) are arbitrary GW approximations of X i , i.e., minimizers of (4), we estimate the relaxed embedded Wasserstein metric as follows:GW 2 (X 1 , (Z, d Z , P 1,♯ π n )) + GW 2 (X 1 , (Z, d Z , P 2,♯ π n )) ≤ 1 λ n EW 2 λn (X 1 , X 2 ) ≤ 1 λ n Z×Z d 2 Z (z, z ′ ) dπ n (z, z ′ ) + 2 i=1GW 2 (X i , (Z, d Z , P i,♯ πn )) ≤ M 2 λ n + GWA(X 1 ) + GWA(X 2 ).Taking the limit n → ∞, and exploiting (Beier & Beinert, 2025, Lem. I.1), we notice that the left-hand side becomesGW 2 (X 1 , (Z, d Z , ζ 1 )) + GW 2 (X 2 , (Z, d Z , ζ 2 )). Hence ζ i , i = 1, 2, have to be GW approximations of X i on (Z, d Z ). □Proof of Proposition 3.6 Without loss of generality, assume that π n converges to π ∈ P(Z × Z) weakly; otherwise take a convergent subsequence. Estimating EW λn using the plans πn := (Id, Id) ♯ ζ n , where ζ n ∈ P(Z) is an arbitrary fixed-support barycenter, i.e., a minimizer of (5), we obtainEW 2 λn (X 1 , X 2 ) ≤ λ n 2 i=1 GW 2 (X i , (Z, d Z , P i,♯ πn ) = λ n GWB(ζ).Thus, λ n → 0 implies EW λn (X 1 , X 2 ) → 0 and therefore W 2 (P 1,♯ π n , P 2,♯ π n ) → 0. Due to the stability of the Wasserstein distance (Ambrosio et al., 2005, Prop. 7.1.3.), the limit of π n has the form π = (Id, Id) ♯ ζ for some ζ ∈ P(Z). Moreover, we haveGW 2 (X 1 , (Z, d Z , P 1,♯ π n )) + GW 2 (X 1 , (Z, d Z , P 2,♯ π n )) ≤ 1 λ n EW 2 λn (X 1 , X 2 ) ≤ GWB(X 1 , X 2 ).Taking the limit n → ∞, and exploiting (Beier & Beinert, 2025, Lem. I.1), we notice that the left-hand side becomesGW 2 (X 1 , (Z, d Z , ζ)) + GW 2 (X 1 , (Z, d Z , ζ)). Hence ζ has to be a fixed-support GW barycenter. □Proof of Proposition 3.7 For arbitrary α ∈ P(X 1 × Z 1 × Z 2 × X 2 ), and for π := P Z1×Z2,♯ α and γ i := P Xi×Zi,♯ α, the functional in ( 9) can be estimated byF λ (α) = (X1×Z1×Z2×X2) 2 1 2 d 2 Z (z 1 , z 2 ) + d 2 Z (z ′ 1 , z ′ 2 ) + λ 2 i=1 (d Xi (x i , x ′ i ) -d Z (z i , z ′ i )) 2 dα(x 1 , z 1 , x 2 , x 2 ) dα(x ′ 1 , z ′ 1 , z ′ 2 , x ′ 2 ) = Z1×Z2 d 2 Z (z 1 , z 2 ) dπ(z 1 , z 2 ) + λ 2 i=1 (Xi×Zi) 2 (d Xi (x i , x ′ i ) -d Z (z i , z ′ i )) 2 dγ i (x i , z i ) dγ i (x ′ i , z ′ i ) = Z1×Z2 d 2 Z (z 1 , z 2 ) dπ(z 1 , z 2 ) + λ (G X1,Z (γ 1 ) + G X2,Z (γ 2 )) ≥ EW 2 λ (X 1 , X 2 ).(15)Now let π * and γ * i be solutions of ( 6). Due to the gluing lemma (Villani, 2003, Lem. 7.6), there exists α* ∈ P(X 1 × Z 1 × Z 2 × X 2 ) such that P Z1×Z2,♯ α * = π * and P Xi×Zi,♯ α * = γ * i .For this α * , the inequality in (15) becomes sharp, i.e., F λ (α * ) = EW λ (X 1 , X 2 ). Thus α * solves (9). The other way round, let α * be a solution of (9). As argued in the first part of the proof, the inequality in (15) has to be again sharp. Hence π * := P Z1×Z2,♯ α * and γ * i := P Xi×Zi,♯ α * solve (6). □
B. Numerical Studies on GW, EW, and EW λOur approach enables geometrically meaningful data comparison as we can approximate the EW metric on general metric spaces Z. We validate this in two experiments.Approximation of EW by EW λ for Synthetic Circular Data Consider circular mm-spacesX i = (S 1 , d S 1 , ξ i ) with (Z, d Z ) = (S 1 , d S 1), where (S 1 , d S 1 ) denotes the circle equipped with a circular distance. We set ξ 1 uniformly distributed and ξ 2 according to the density of a von Mises distribution with increasing dispersion parameter κ. Note that the von Mises distribution takes the form of a uniform distribution for κ = 0 and becomes more concentrated for κ → ∞. Due to the rotational invariance of the uniform distribution on the circle, the embedded Wasserstein distance can easily be calculated in this case as it coincides with the Wasserstein distance. Thus, we can ignore the isometry in EW and directly calculate EW by solving the linear program underlying the OT problem. We discretize the circle into 360 bins and estimate EW λ for different choices of κ and λ. The results in Figure 8 show an improved approximation of EW for larger λ as announced in Proposition 3.4. Indeed, we have an excellent fit for λ = 20, whereas λ = 0.2 leads to an underestimation.2d Shape Matching Next, we compare randomly rotated gray-value images. We can describe such images as mm-spaces X i = (X, d, ξ i ) where X ⊂ R 2 is a grid that describes the pixel positions equipped with the Euclidean metric and ξ i is the pixel intensity, see (Beier et al., 2023). We use (Z, d Z ) = (X, d). We apply random affine transformations, i.e., translations and rotations, to the first 10 FashionMNIST (Xiao et al., 2017) training images from the classes "Trouser", "Pullover" and "Sneaker", respectively. Then we compute W, GW and EW λ for λ = 20. The results are displayed in Figure 9. As expected, the Wasserstein distance cannot recover the class structure due to the affine transformations, whereas EW λ and GW capture the class structure. Moreover, we see that the distance matrices of EW λ and GW display almost identical patterns, up to scaling.  EW 2 (λ = ∞) EW 2 λ (λ = 20) EW 2 λ (λ = 0.2)
C. Parameter SensitivityAlgorithm 1 depends on three input parameters, namely the GW regularization parameter λ, the entropic regularization parameter ε and the chosen discrete reference space Z. Generally, we aim to set λ as high as possible and ε as low as possible while preserving numerical stability. Based on two sets of 3d shapes described in Section 6.1, namely a s-curve in combination with a Swiss roll with a hole and two human shapes, we present a parameter sensitivity study that illustrates the impact of λ, ε and Z. Throughout this study note that our Euclidean joint embeddings are invariant to translations and rotations which results in arbitrarily oriented joint embeddings. In Figure 10, we investigate the influence of the GW parameter λ that promotes near-isometric embeddings. We observe that the embeddings of both spaces nearly coincide for small λ. This is supported by Proposition 3.6 showing that the embeddings form a fixed-support barycenter in the limiting case. Moreover, we see a saturation effect where an increase of λ from 1 to 100 has only a small impact on the joint embeddings. In Figure 11, we investigate the influence of the entropic parameter ε that enables efficient computation. Here, we see that a large ε leads to a highly blurred embedding. This shows that it is advisable to choose a low, but numerically stable value for ε. Lastly, we investigate the discretization of the reference space in Figure 12. Again, we aim for a fine discretization, but the runtime of our algorithm depends heavily on the size |Z| of our reference space. We observe that the joint embeddings display the same shapes and alignments, only at higher resolution. All embeddings were computed with 40 iterations of Algorithm 1.publicly available SCOT implementation3 to estimate a suitable nearest-neighbor graph. Here, we use the same ε as in the rest of the experiment and use the default grid over the neighborhood size. Note that the public implementation uses a connectivity graph based on pairwise correlation distances between points instead of Euclidean distances, see (Demetci et al., 2022). Then, we use SCOT and MDS to estimate the final embeddings based on the pairwise geodesic distance matrix estimated from this graph. We use the same distance graph for UnionCom, JMDS, and our algorithm. Here, we estimate suitable hyperparameters based on a random 10% validation split. We use four parameter configurations for each algorithm, i.e., we test λ = 0.01, 0.1, 10, 100 for JMDS and our method. For UnionCOM, we use combinations of β = 1, 10 and a perplexity of 30 and 100 for the t-SNE dimensionality reduction. We use no parameter annealing for any method.
E. Further Example for Comparing Feature SpacesAs an addition to the genetyping experiment, we present another experiment with real-world data based on latent spaces. We consider the task of comparing the 4d latent spaces of an auto-encoder (AE) versus a variational auto-encoder (VAE), trained on FashionMNIST (Xiao et al., 2017). For this purpose, we train a simple convolutional AE with two convolutional layers and one linear layer for the encoder and the decoder. We train the AE with the MSE loss and the VAE with the log-likelihood loss for 10 epochs on the canonical training data splits. Subsequently, we embed the test data using the resulting AE and VAE into the 4d latent spaces. Note that such distinct AE trained on the same dataset generally produce incomparable embeddings. For our experiment, we consider the first 100 data points of the test split. We choose Z as an equispaced 20×20 grid on [0, 1.3] ⊂ R 2 with the Euclidean distance. We follow the same evaluation routine as in Subsection 6.2 for parameter selection and visualization. The joint embeddings are visualized in Figure 13. Comparing FOSCTTM and KNN, we see again a good embedding quality for our approach, see the figure caption. Figure1. Joint (aligning) transfer of two metric spaces (gray surfaces with surface distance) to a fixed reference space, namely to the sphere and the torus by our method, where the color "yellow" corresponds to higher values, see Subsection 6.1.
Figure 2 .2Figure 2. Illustration of GW formulations of Sturm and Mémoli.
Figure 3 .3Figure 3. Illustration of our multi-marginal transport problem. The colors are in line with Figure 2: quantities with respect to X1 are red, and quantities related to X2 are blue.
S-bended rectangle and Swiss roll with hole.
Figure 4 .4Figure 4. Embedding and alignment of an S-bended rectangle and a Swiss roll into R 2 .For our method, we visualize the marginals P Z i ,♯ α of the computed 4-plan α in (9) and compare them with JMDS. In the second example (b), JMDS produces an unexpected hole when embedding the S-bended surface.
are not possible. More precisely, we embed human shapes from the FAUST dataset(Bogo et al., 2014) into R 2 . For this, we choose (Z, d Z ) as an equispaced 60×60 grid on [0, 1.3] 2 ⊂ R 2 and apply Algorithm 1 with λ = 100 and ε = 4 • 10 -4 . The obtained transport-based embeddings are shown in Figure5. As comparison, we apply JMDS with λ = 10 and ε = 4 • 10 -4 . The 2d representations clearly resemble the 3d human shapes, but JMDS splits some of the extremities. Since our method and JMDS can both be derived from the same functional, we additionally compare the achieved objectives, the Wasserstein distances between embeddings, and the GW distances between embeddings and input spaces. The experiment is repeated for each pairwise combination of the first ten shapes from the FAUST dataset and different λ. For comparison, the shapes are GW 2W 2
Figure 6 .6Figure6. Joint embedding of two feature spaces into R 2 using our method and JMDS (Top: first feature space, Bottom: second feature space). Both methods align the color-coded classes.
consists of d 1 = 19 and d 2 = 10 features of n = 1047 single cells, and scGEM of d 1 = 34 and d 2 = 27 features of n = 177 specimens.
Figure 7 .7Figure 7. Alignment of GMMs with respect to the Wasserstein distance. The considered and computed GMMs are visualized via their density function on R 2 .
Figure 8 .Figure 9 .89Figure 8. Comparison of EW with EW λ for different λ. While the curves for EW (left) and EW20 (middle) are almost identical, those for EW0.2 (right) is consistently smaller.
Figure 10 .10Figure 10. Ablation study of λ for joint embedding of shapes from Figure 4b and 5a. Based on varying λ values, we employ our method with ε = 0.001 and Z defined by a uniform 20 × 20 grid in [0, 1.3] 2 .
Figure 13 .13Figure13. Joint embedding of the FashionMNIST latent space of an AE and a VAE into R 2 using our method and JMDS. The AE latent space is on the top and the VAE latent space is on the bottom. Both methods align the color-coded classes. A quantitative comparison shows that our model achieves a better FOSCTTM (0.040) than JMDS (0.042), whereas our KNN-Accuracy (0.589) is slightly worse than the one of JMDS (0.644).
Table 2 .2Comparison of the joint embedding of two feature spaces into R 2 using our method, JMDS, SCOT, and UnionCom.
			https://github.com/MoePien/ RelaxedEmbeddedWasserstein
			https://rsinghlab.github.io/SCOT/data/
			https://github.com/rsinghlab/SCOT(March 2025)