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Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization

T0 review · 2 major / 7 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper proposes an unsupervised method to align two heterogeneous datasets in a common metric space via unbalanced optimal transport with Gromov–Wasserstein marginal penalization, and proves convergence to the embedded Wasserstein…

desk verdict Solid theoretical contribution on unbalanced OT with GW marginal penalization, but the numerical section rests on an unproven relaxation gap that should be addressed before the numbers are taken at face value. read the letter →

arxiv 2502.07510 v2 pith:E3YKYHJ5 submitted 2025-02-11 cs.LG

Florian Beier , Moritz Piening , Robert Beinert , Gabriele Steidl This is my paper · ORCID
classification cs.LG MSC 49Q22
keywords jointmetricspaceembeddingunbalancedoptimaltransportGromov-WassersteindistanceembeddedWassersteinmulti-marginalbi-convexrelaxationmeasurespaces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Suppose two datasets live in different spaces and no point correspondences are known. This paper proposes to map both into a fixed reference metric space by minimizing an unbalanced optimal transport plan between the two embedded copies (the marginals are penalized rather than prescribed), while adding a Gromov–Wasserstein penalty on each marginal that rewards preserving the internal distances of the original datasets. The main theoretical results are that a minimizer always exists and that, as the penalty parameter goes to infinity, minimizers converge to the embedded Wasserstein distance when isometric embeddings exist and to best-approximating marginals otherwise. A reformulation as a quadratic, multi-marginal, unbalanced optimal transport problem yields a bi-convex relaxation solvable by alternating Sinkhorn updates, and the authors demonstrate joint embeddings in Euclidean space, on spheres and tori, and in a space of Gaussians. The practical payoff is an unsupervised way to align, compare, and transfer labels between heterogeneous datasets whose only common ground is the pairwise distances within each dataset.

What carries the argument

The central object is the functional $EW_\lambda$ in (3): $EW_\lambda(X_1,X_2)=\inf_{\pi\in\mathcal{P}(Z\times Z)}\left(\int_{Z\times Z} d_Z^2(z,z')\,d\pi(z,z')+\lambda\sum_{i=1}^2 GW^2(X_i,(Z,d_Z,P_{i\#}\pi))\right)^{1/2}$. The first term is an ordinary transport cost between the two embedded marginals; each Gromov–Wasserstein term pulls its marginal toward an isometric copy of its source space. The paper rewrites this as a quadratic, multi-marginal, unbalanced optimal transport problem over a 4-plan $\alpha\in\mathcal{P}(X_1\times Z_1\times Z_2\times X_2)$ and relaxes it to the bilinear objective $F_\lambda(\alpha_1,\alpha_2)$; Algorithm 1 alternates multi-marginal Sinkhorn steps in the two factors. This fixed-support discretization is what allows the method to run in arbitrary metric spaces rather than only Euclidean ones.

What would settle it

One concrete check: on a small discrete pair of metric spaces, compute the true quadratic minimum of (9) by exhaustive enumeration and run Algorithm 1 from several random initializations; if the final objectives differ or stay above the enumerated minimum, the bi-convex relaxation is not reliably solving the original problem.

Watch

Extended reading notes

Core claim

The paper's central claim is that the functional $EW_\lambda$ in (3) — an unbalanced optimal transport cost with Gromov–Wasserstein marginal penalization — is a well-posed relaxation of the embedded Wasserstein distance $EW$. Proposition 3.3 proves that a minimizer exists for every $\lambda>0$. Proposition 3.4 shows that any sequence of minimizers with $\lambda_n\to\infty$ has a weakly convergent subsequence whose limit realizes $EW$ through isometries whenever the source spaces admit isometric embeddings into the reference space; Proposition 3.5 covers the non-isometric case, where the limit marginals are Gromov–Wasserstein approximations; Proposition 3.6 gives the $\lambda\to0$ limit as a fixed-support Gromov–Wasserstein barycenter. Proposition 3.7 establishes the equivalence between this formulation and the 4-plan quadratic problem (9), which is the basis of the numerical method.

Load-bearing premise

The numerical results assume that the alternating Sinkhorn updates on the relaxed problem land near the true quadratic optimum; the paper proves neither that the two factors converge to the same plan nor that the relaxation gap is small.

Editorial extensions

If this is right

  • For any compact reference space and any $\lambda>0$, the optimization problem is well-posed: $EW_\lambda$ has a minimizer, so the method does not rely on an external feasibility argument.
  • As $\lambda$ grows, the alignments approach the exact embedded Wasserstein alignment whenever the source spaces can be isometrically embedded; when they cannot, the limit marginals are the best Gromov–Wasserstein approximations available in the reference space.
  • Because the support of the embedded measures is fixed in advance, the method works for arbitrary metric spaces such as spheres, tori, and the Wasserstein space of Gaussians, not only Euclidean space.
  • As $\lambda\to0$, the limit is a fixed-support Gromov–Wasserstein barycenter, connecting joint embedding to metric-measure-space barycenter computation.
  • The same objective, with free supports and fixed weights, reduces to joint multidimensional scaling, so the framework covers both fixed-support and free-support joint embeddings in one model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The convergence guarantees apply to minimizers, not to whatever point the alternating Sinkhorn solver reaches; a natural stress test is to run Algorithm 1 from several initializations on a small instance and compare the final objective with the true quadratic minimum found by enumeration.
  • Because the marginal-penalization idea does not depend on there being exactly two sources, a chained Wasserstein term between consecutive embeddings is a direct way to extend the model to several datasets; the paper notes the multi-marginal Sinkhorn step scales linearly in the number of spaces.
  • Replacing the fixed grid by optimized free supports would remove the current restriction to low-dimensional reference grids and would connect the approach to free-support non-Euclidean barycenter solvers.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The paper introduces a new unsupervised alignment framework for two metric-measure spaces into a fixed compact metric space (Z, dZ). The proposed functional EW_lambda combines a Wasserstein cost on the joint embedding with Gromov-Wasserstein penalty terms on the marginals. The authors prove existence of minimizers (Proposition 3.3), characterize the limits as the penalization parameter tends to infinity (Propositions 3.4 and 3.5) and to zero (Proposition 3.6), and reformulate the problem as a quadratic multi-marginal unbalanced optimal transport problem (9). A bi-convex relaxation (10) is solved numerically by block-coordinate descent with an entropic multi-marginal Sinkhorn scheme, and experiments are reported for Euclidean and non-Euclidean target spaces, including shape embedding, single-cell feature alignment, and Gaussian mixture alignment.

Significance. If the theoretical results are correct, the paper provides a principled extension of Wasserstein Procrustes and JMDS to arbitrary compact metric spaces, with rigorous links between a computable relaxation and the embedded Wasserstein distance. The existence and lambda-limit proofs in Appendix A use standard compactness and lower semicontinuity arguments and are largely coherent; the derivation of the embedded Wasserstein distance as the infinite-penalization limit is a genuine theoretical contribution. The numerical experiments cover diverse settings, and the authors provide a public code repository. However, the significance of the paper as a computational method depends on the unresolved gap between the quadratic problem (9) and the bilinear relaxation (10) actually solved by Algorithm 1, which is not analyzed in the manuscript.

major comments (2)
  1. [Section 4, Eq. (10) and Algorithm 1] The numerical viability of the method is the paper's third stated contribution, but it rests on an unproven equivalence between the quadratic problem (9) and the bilinear relaxation (10) as actually solved. Equation (10) is only a lower bound on (9), and the text observes only that a bi-convex minimizer with alpha1 = alpha2 solves the original problem; it does not show that the alternating Sinkhorn updates in Algorithm 1 converge to such a coupled solution, nor does it bound the gap between the objective of the computed 4-plan and EW_lambda. Block-coordinate descent on a bilinear objective can stall at saddle points or cycles with alpha1 != alpha2, and each Sinkhorn step solves an epsilon-regularized linear subproblem, adding an unquantified perturbation. Consequently, the values and comparisons reported in Section 6 (Tables 1-2, Figures 4-7) are not certified as approximate minima of EW_lambda; they could be minima of a strictly looser problem. I consider this load-bearing because the abstract and contribution list explicitly present the bi-convex relaxation as the computational route to the model.
  2. [Section 5 and Section 6.1] The quantitative comparison with JMDS is not fully controlled. JMDS optimizes free supports in Euclidean space under fixed uniform weights, while EW_lambda fixes the grid Z and optimizes weights; moreover, in the figures the two methods use different values of lambda and epsilon (e.g., Figure 4 and Figure 5 use lambda = 100 for the proposed method and lambda = 10 for JMDS). The statement in Section 6.1 that 'we here achieve better joint embeddings than JMDS' is therefore a claim about the particular discretizations and regularizations, not a clean comparison of the underlying objectives. The paper should either match the hyperparameters and computational budgets more carefully, or explicitly state that the comparison is between a fixed-support weight-optimization formulation and a free-support formulation under their respective recommended settings.
minor comments (7)
  1. [Proposition 3.7] The statement of Proposition 3.7 is not well-formed as printed: it reads 'If α solves (3), then its projections (3) are solutions of (3).' The first occurrence should refer to the 4-plan problem (9), and the second to the projections defined in (8). The mislabelled references should be corrected.
  2. [Section 4, first paragraph] The sentence 'the computation of EWλ in (3) requires the solution of a quadratic optimization problem' should refer to the reformulation (9), not to the original functional (3).
  3. [Proof of Proposition 3.4] The display 'EWλn = ∫ d^2 + λ ∑ GW^2' omits the square root from the definition in (3); the chain of inequalities that follows is harder to follow as a result, and the notation should be made consistent.
  4. [Proof of Proposition 3.2] There are typographical errors: 'we we show' and 'isometies' should read 'we show' and 'isometries'.
  5. [Section 5 and Table 2] There are small typos: 'JMSD' should be 'JMDS' in Section 5, and 'KKN-Acc' in Table 2 should be 'KNN-Acc'.
  6. [Figure 12 caption] The notation |Z| = 15^2, 20^2, 60^2 is ambiguous; it should state explicitly that these are 15×15, 20×20, and 60×60 grids, respectively.
  7. [Figure 4 caption and text] The text refers to the 'Swiss role' in one place; this should be 'Swiss roll'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence theorems are proven from compactness and lower semi-continuity, and the numerical relaxation is explicitly labeled a lower bound.

full rationale

The central theoretical results are self-contained rather than definitionally forced. EWλ in (3) is a genuinely penalized variant of EW in (2), and Propositions 3.3–3.6 establish existence and λ→∞/λ→0 limits using weak compactness of P(Z×Z), joint weak lower semi-continuity of the Wasserstein term, and lower semi-continuity of the GW marginal terms; the key estimate EWλ ≤ EW in (14) is an inequality to be proved, not an equality by construction, and the reverse bound is obtained from lower semi-continuity. The equivalent 4-plan reformulation (9) and the bi-convex relaxation (10) are explicitly identified: 'By construction, the minimizers of (4) constitute a lower bound to the original, quadratic problem (3)', and Proposition 3.7 only relates exact minimizers of the original and relaxed problems. The paper nowhere claims that the Sinkhorn iterates certify a minimum of EWλ; the lack of a convergence or gap bound for Algorithm 1 is an unproven numerical assumption, which is a correctness risk rather than a circular step. Self-citations to Beier et al. (2022) for the multi-marginal Sinkhorn scheme and to Beier & Beinert (2025, Lem. I.1) for GW lower semi-continuity are technical building blocks; the lemma's assumptions are weak convergence on compact metric spaces and do not include the paper's convergence conclusions, and the same lower semi-continuity argument is essentially developed in Proposition 3.3. No fitted parameter is renamed as a prediction, and no uniqueness or existence theorem is imported solely from the authors' prior work to force the choice of model. Therefore no circular step can be exhibited.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model introduces no new physical entities. It rests on standard compactness and lower semicontinuity results plus the metric property of GW. The computational results depend on user-chosen hyperparameters lambda, epsilon, and the target grid Z.

free parameters (4)
  • lambda (GW penalization weight) = 1, 10, 100, 1000 in experiments; grid-searched in Section 6.2
    Trades off geometry preservation against Wasserstein matching; the convergence theorems hold for any lambda > 0, but numerical results depend on its choice.
  • epsilon (entropic regularization) = 10^-3 or 4*10^-4
    Chosen small to avoid overflow; larger epsilon blurs the embeddings and changes the objective.
  • Target space grid Z = 15x15 to 60x60 grids
    The method requires a fixed finite metric space; grid resolution controls runtime and embedding quality.
  • r^2 (GMM variance scale) = chosen per dataset
    In Section 6.3, the covariance grid for the target space is scaled by the mean variance of the GMMs, a hand-chosen scale.
assumptions (5)
  • standard math Xi and Z are compact metric spaces and measures are Borel probability measures.
    Invoked throughout Section 3 and Appendix A to use weak compactness and Weierstrass theorem.
  • standard math Wasserstein and GW distances are jointly weakly lower semicontinuous.
    Used in Propositions 3.3-3.6; cited to Ambrosio, Gigli, Savare and established GW theory.
  • domain assumption GW(X,Y)=0 implies a measure-preserving isometry between the supports.
    Needed in Proposition 3.4 to extract isometries from vanishing GW penalty; standard property of GW as a metric on isomorphism classes of mm-spaces.
  • standard math Gluing lemma for couplings.
    Used in Proposition 3.7 to assemble the 4-plan from pi and gamma_i.
  • domain assumption Lemma I.1 of Beier and Beinert 2025 on stability of GW barycenters.
    Self-cited in proofs of Propositions 3.5 and 3.6 to pass to limits of GW approximation functionals.

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Cite this review

Pith. "Pith review of Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization." pith.science (2026). https://pith.science/paper/E3YKYHJ5

@misc{pith2026250207510,
  author       = {Pith},
  title        = {Pith review of: Joint Metric Space Embedding by Unbalanced OT with Gromov-Wasserstein Marginal Penalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3YKYHJ5}},
  note         = {Machine review of arXiv:2502.07510}
}
read the original abstract

We propose a new approach for unsupervised alignment of heterogeneous datasets, which maps data from two different domains without any known correspondences to a common metric space. Our method is based on an unbalanced optimal transport problem with Gromov-Wasserstein marginal penalization. It can be seen as a counterpart to the recently introduced joint multidimensional scaling method. We prove that there exists a minimizer of our functional and that for penalization parameters going to infinity, the corresponding sequence of minimizers converges to a minimizer of the so-called embedded Wasserstein distance. Our model can be reformulated as a quadratic, multi-marginal, unbalanced optimal transport problem, for which a bi-convex relaxation admits a numerical solver via block-coordinate descent. We provide numerical examples for joint embeddings in Euclidean as well as non-Euclidean spaces.

Figures

Figures reproduced from arXiv: 2502.07510 by the authors.

Figure 1
Figure 1. Joint (aligning) transfer of two metric spaces (gray sur￾faces 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. of such heterogeneous settings are, e.g., the comparison of graph- or mesh-valued data such as 3d shapes or mani￾folds. This paper introduces a novel framework based on OT which en… view at source ↗
Figure 2
Figure 2. Illustration of GW formulations of Sturm and Memoli. ´ For this specific setting, we relax Sturm’s GW distance to EW(X1, X2) := inf I1 : supp ξ1,→Z I2 : supp ξ2,→Z W(I1,♯ξ1, I2,♯ξ2). (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 X1, X2, and Z, this reduces to the Wasserstein Procrustes problem… view at source ↗
Figure 3
Figure 3. Illustration of our multi-marginal transport problem. The colors are in line with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Embedding and alignment of an S-bended rectangle and a Swiss roll into R 2 . For our method, we visualize the marginals PZi,♯α of the computed 4-plan α in (3) and compare them with JMDS. In the second example (b), JMDS produces an unexpected hole when embedding the S-b…
Figure 5
Figure 5. Figure 5: Embedding and alignment of human shapes from the FAUST dataset into R 2 . For our method, we visualize the marginals PZi,♯α of the computed 4-plan α in (3) and compare them with JMDS. Here JMDS tends to split some of the extremities. As comparison, we also show the res…
Figure 6
Figure 6. Figure 6: 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. SNAREseq scGEM FOSCTTM↓ KNN-Acc↑ Time FOSCTTM↓ KNN-Acc↑ Time EWλ (Ours) 0.165 0.943 28.9s 0.20…
Figure 7
Figure 7. Figure 7: Alignment of GMMs with respect to the Wasserstein distance. The considered and computed GMMs are visualized via their density function on R 2 . GW barycenter algorithms (Peyre et al. ´ , 2016; Vayer et al., 2020a; Beier & Beinert, 2025). This may combine our EWλ approa…
Figure 8
Figure 8. Figure 8: Comparison of EW with EWλ for different λ. While the curves for EW (left) and EW20 (middle) are almost identical, those for EW0.2 (right) is consistently smaller. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Pairwise W, GW and EW20 distances of 10 randomly rotated and translated images from each of the “Trouser” (T), “Pullover” (P), and “Sneaker” (S) FashionMNIST classes. Smaller distances are darker. The class block structure is recovered with EWλ and GW, but not with Was…
Figure 10
Figure 10. Figure 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 . C. Parameter Sensitivity Algorithm 1 depends on three input parameters, namel…
Figure 11
Figure 11. Figure 11: Ablation study of ε for joint embedding of shapes from Figure 4b and 5a. Based on varying ε values, we employ our method with λ = 100 and Z defined by a uniform 20 × 20 grid in [0, 1.3]2 . Below the joint embeddings, we show convergence plots of the unregularized func…
Figure 12
Figure 12. Figure 12: Ablation study of the grid size for joint embedding of shapes from Figure 4b and 5a. Based on ε = 0.001 and λ = 100, we employ our method with Z ⊂ [0, 1.3]2 defined by a uniform 15 × 15, 20 × 20 or 60 × 60 grid. We include CPU runtimes. D. Hyperparameter Selection JMD…
Figure 13
Figure 13. Figure 13: 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 mo…

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Works this paper leans on

46 extracted references · 43 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

  2. [2]

    Z., B \'e rar, M., Gasso, G., and Rakotomamonjy, A

    Alaya, M. Z., B \'e rar, M., Gasso, G., and Rakotomamonjy, A. Theoretical guarantees for bridging metric measure embedding and optimal transport. Neurocomputing, 468: 0 416--430, 2022

    work page 2022

  3. [3]

    Alvarez-Melis, D., Jegelka, S., and Jaakkola, T. S. Towards optimal transport with global invariances. In International Conference on Artificial Intelligence and Statistics (AISTATS), volume 89 of Proceedings of Machine Learning Research, pp.\ 1870--1879. PMLR, 2019

    work page 2019

  4. [4]

    Gradient Flows in Metric Spaces and in the Space of Probability Measures

    Ambrosio, L., Gigli, N., and Savar\'e, G. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkh\"auser, Basel, 2005

    work page 2005

  5. [5]

    and Krishnaswamy, S

    Amodio, M. and Krishnaswamy, S. MAGAN : Aligning biological manifolds. In International Conference on Machine Learning (ICML), pp.\ 215--223. PMLR, 2018

    work page 2018

  6. [6]

    and Beinert, R

    Beier, F. and Beinert, R. Tangential fixpoint iterations for Gromov -- Wasserstein barycenters. SIAM J. Imaging Sci., 18 0 (2): 0 1058--1100, 2025

    work page 2025

  7. [7]

    Unbalanced multi-marginal optimal transport

    Beier, F., von Lindheim, J., Neumayer, S., and Steidl, G. Unbalanced multi-marginal optimal transport. J. Math. Imaging Vision, 65: 0 394--413, 2022

    work page 2022

  8. [8]

    Multi-marginal G romov-- W asserstein transport and barycentres

    Beier, F., Beinert, R., and Steidl, G. Multi-marginal G romov-- W asserstein transport and barycentres . Inf. Inference, 12 0 (4): 0 2753--2781, 10 2023. ISSN 2049-8772

    work page 2023

Show all 46 references
  1. [9]

    Bogachev, V. I. Weak convergence of measures. Number 234 in Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2018

    2018

  2. [10]

    Bogo, F., Romero, J., Loper, M., and Black, M. J. FAUST : Dataset and evaluation for 3D mesh registration. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, June 2014

    2014

  3. [11]

    Unsupervised topological alignment for single-cell multi-omics integration

    Cao, K., Bai, X., Hong, Y., and Wan, L. Unsupervised topological alignment for single-cell multi-omics integration. Bioinformatics, 36 0 (1): 0 i48--i56, 2020

    2020

  4. [12]

    Manifold alignment for heterogeneous single-cell multi-omics data integration using Pamona

    Cao, K., Hong, Y., and Wan, L. Manifold alignment for heterogeneous single-cell multi-omics data integration using Pamona . Bioinformatics, 38 0 (1): 0 211--219, 2022

    2022

  5. [13]

    Carroll, J. D. and Arabie, P. Multidimensional scaling. In Birnbaum, M. H. (ed.), Measurement, Judgment and Decision Making, pp.\ 179--250. Elsevier, 1998

    1998

  6. [14]

    Unsupervised manifold alignment with joint multidimensional scaling

    Chen, D., Fan, B., Oliver, C., and Borgwardt, K. Unsupervised manifold alignment with joint multidimensional scaling. In International Conference on Learning Representations (ICLR). OpenReview, 2023

    2023

  7. [15]

    B., and Zhang, K

    Chen, S., Lake, B. B., and Zhang, K. High-throughput sequencing of the transcriptome and chromatin accessibility in the same cell. Nat. Biotechnol., 37: 0 1452--1457, 2019

    2019

  8. [16]

    F., Courtois, E

    Cheow, L. F., Courtois, E. T., Tan, Y., Viswanathan, R., Xing, Q., Tan, R. Z., Tan, D. S. W., Robson, P., Loh, Y.-H., Quake, S. R., and Burkholder, W. F. Single-cell multimodal profiling reveals cellular epigenetic heterogeneity. Nat. Methods, 13: 0 833--836, 2016

    2016

  9. [17]

    A., Needham, T., and Weighill, T

    Clark, R. A., Needham, T., and Weighill, T. Generalized dimension reduction using semi-relaxed G romov- W asserstein distance. In AAAI Conference on Artificial Intelligence (AAAI), volume 39, pp.\ 16082--16090. AAAI Press, 2025

    2025

  10. [18]

    Generalized unsupervised manifold alignment

    Cui, Z., Chang, H., Shan, S., and Chen, X. Generalized unsupervised manifold alignment. In Advances in Neural Information Processing Systems (NeurIPS), volume 27. Curran Associates, Inc., 2014

    2014

  11. [19]

    and Avis, D

    Cuturi, M. and Avis, D. Ground metric learning. JMLR, 15 0 (1): 0 533--564, 2014

    2014

  12. [20]

    S., and Singh, R

    Demetci, P., Santorella, R., Sandstede, B., Noble, W. S., and Singh, R. SCOT : single-cell multi-omics alignment with optimal transport. J. Comput. Biol., 29 0 (1): 0 3--18, 2022

    2022

  13. [21]

    Neuc- MDS : Non- E uclidean multidimensional scaling through bilinear forms

    Deng, C., Gao, J., Lu, K., Luo, F., Sun, H., and Xin, C. Neuc- MDS : Non- E uclidean multidimensional scaling through bilinear forms. In Advances on Neural Information Processing Systems (NeurIPS), volume 38. Curran Associates, Inc., 2024

    2024

  14. [22]

    Dijkstra, E. W. A note on two problems in connexion with graphs. Numer. Math., 1: 0 269--271, 1959

    1959

  15. [23]

    Statistical optimal transport via factored couplings

    Forrow, A., H \"u tter, J.-C., Nitzan, M., Rigollet, P., Schiebinger, G., and Weed, J. Statistical optimal transport via factored couplings. In International Conference on Artificial Intelligence and Statistics (AISTATS), Proceedings of Machine Learning Research, pp.\ 2454--24...

    2019

  16. [24]

    Unsupervised alignment of embeddings with W asserstein procrustes

    Grave, E., Joulin, A., and Berthet, Q. Unsupervised alignment of embeddings with W asserstein procrustes. In Chaudhuri, K. and Sugiyama, M. (eds.), International Conference on Artificial Intelligence and Statistics (AISTATS), volume 89 of Proceedings of Machine Learning Resear...

    2019

  17. [25]

    J., Hastie, T., d’Enza, A

    Greenacre, M., Groenen, P. J., Hastie, T., d’Enza, A. I., Markos, A., and Tuzhilina, E. Principal component analysis. Nat. Rev. Methods Primers, 2 0 (1): 0 100, 2022

    2022

  18. [26]

    Ground metric learning on graphs

    Heitz, M., Bonneel, N., Coeurjolly, D., Cuturi, M., and Peyr \'e , G. Ground metric learning on graphs. J. Math. Imaging Vision, 63: 0 89--107, 2021

    2021

  19. [27]

    P., Welling, M., et al

    Kingma, D. P., Welling, M., et al. An introduction to variational autoencoders. Found. Trends Theor. Comput. Sci., 12 0 (4): 0 307--392, 2019

    2019

  20. [28]

    Liu, J., Huang, Y., Singh, R., Vert, J.-P., and Noble, W. S. Jointly embedding multiple single-cell omics measurements. In 19th International Workshop on Algorithms in Bioinformatics (WABI 2019), volume 143 of Leibniz International Proceedings in Informatics (LIPIcs), pp.\ 10:...

    2019

  21. [29]

    UMAP : Uniform manifold approximation and projection

    McInnes, L., Healy, J., Saul, N., and Gro berger, L. UMAP : Uniform manifold approximation and projection. J. Open Source Softw., 3 0 (29), 2018

    2018

  22. [30]

    G romov-- W asserstein distances and the metric approach to object matching

    M \'e moli, F. G romov-- W asserstein distances and the metric approach to object matching. Found. Comput. Math., 11 0 (4): 0 417--487, 2011

    2011

  23. [31]

    Mordukhovich, B. S. and Nam, N. M. Convex analysis and beyond. Springer, 2022

    2022

  24. [32]

    Topology

    Munkres, J. Topology. Featured Titles for Topology. Prentice Hall, Upper Saddle River, 2000

    2000

  25. [33]

    and Cuturi, M

    Paty, F.-P. and Cuturi, M. Subspace robust W asserstein distances. In International Conference on Machine Learning (ICML), pp.\ 5072--5081. PMLR, 2019

    2019

  26. [34]

    Scikit-learn: Machine learning in P ython

    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., et al. Scikit-learn: Machine learning in P ython. JMLR, 12: 0 2825--2830, 2011

    2011

  27. [35]

    G romov-- W asserstein averaging of kernel and distance matrices

    Peyr\'e, G., Cuturi, M., and Solomon, J. G romov-- W asserstein averaging of kernel and distance matrices. In Balcan, M. F. and Weinberger, K. Q. (eds.), International Conference on Machine Learning (ICML), volume 48 of Proceedings of Machine Learning Research, pp.\ 2664--2672...

    2016

  28. [36]

    Roweis, S. T. and Saul, L. K. Nonlinear dimensionality reduction by locally linear embedding. Science, 290 0 (5500): 0 2323--2326, 2000

    2000

  29. [37]

    Gromov -- Wasserstein -like distances in the Gaussian mixture models space

    Salmona, A., Desolneux, A., and Delon, J. Gromov -- Wasserstein -like distances in the Gaussian mixture models space. TMLR, 2024

    2024

  30. [38]

    On the geometry of metric measure spaces

    Sturm, K.-T. On the geometry of metric measure spaces . Acta Mathematica, 196 0 (1): 0 65 -- 131, 2006

    2006

  31. [39]

    The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces, volume 290 of Memoirs of the American Mathematical Society

    Sturm, K.-T. The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces, volume 290 of Memoirs of the American Mathematical Society. American Mathematical Society, Providence, 2023

    2023

  32. [40]

    The unbalanced G romov-- W asserstein distance: Conic formulation and relaxation

    Séjourné, T., Vialard, F.-X., and Peyr\'e, G. The unbalanced G romov-- W asserstein distance: Conic formulation and relaxation. In Ranzato, M., Beygelzimer, A., Dauphin, Y., Liang, P., and Vaughan, J. W. (eds.), Advances in Neural Information Processing Systems (NeurIPS), volu...

    2021

  33. [41]

    Distributional reduction: Unifying dimensionality reduction and clustering with G romov- W asserstein

    Van Assel, H., Vincent-Cuaz, C., Courty, N., Flamary, R., Frossard, P., and Vayer, T. Distributional reduction: Unifying dimensionality reduction and clustering with G romov- W asserstein. arXiv preprint arXiv:2402.02239, 2024

    2024 arXiv

  34. [42]

    and Hinton, G

    Van der Maaten, L. and Hinton, G. Visualizing data using t- SNE . JMLR, 9 0 (86): 0 2579--2605, 2008

    2008

  35. [43]

    Fused Gromov -- Wasserstein distance for structured objects

    Vayer, T., Chapel, L., Flamary, R., Tavenard, R., and Courty, N. Fused Gromov -- Wasserstein distance for structured objects. Algorithms, 13 0 (9): 0 212, 2020 a

    2020

  36. [44]

    Co-optimal transport

    Vayer, T., Redko, I., Flamary, R., and Courty, N. Co-optimal transport. In Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., and Lin, H. (eds.), Advances in Neural Information Processing Systems (NeurIPS), volume 33 of Advances in Neural Information Processing Systems, pp....

    2020

  37. [45]

    Topics in optimal transportation

    Villani, C. Topics in optimal transportation. Number 58 in Graduate Studies in Mathematics. American Mathematical Society, Procidence, 2003

    2003

  38. [46]

    Fashion-MNIST : A novel image dataset for benchmarking machine learning algorithms

    Xiao, H., Rasul, K., and Vollgraf, R. Fashion-MNIST : A novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017

    2017 arXiv

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.