{"id":"1e3d8ea3-5f80-442c-82c0-67c7c7af3ff5","arxiv_id":"2502.07510","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A new unbalanced optimal transport formulation with Gromov-Wasserstein marginal penalties computes joint embeddings of heterogeneous datasets into a common metric space and provably converges to the embedded Wasserstein distance as the penalty grows.","lead":"Researchers introduce a new optimal transport method that maps two datasets without known correspondences into a shared metric space, balancing how much each dataset's internal geometry is preserved with how well the embedded versions match. The approach generalizes Euclidean alignment techniques to arbitrary metric spaces such as spheres, tori, and spaces of Gaussian distributions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Numerical claims rest on an unproven tightness/convergence guarantee for the bilinear relaxation (10); block-coordinate descent may not produce a coupled solution, and no gap bound is given.","rationale":"After checking the main existence and limit-type arguments (Propositions 3.3-3.6), I find no flaw that would invalidate the theoretical contribution: weak compactness plus lower semicontinuity gives existence, and the lambda -> infinity and lambda -> 0 arguments use boundedness by EW or GWB together with the same GW lower semicontinuity, so those claims stand. The single load-bearing weakness is the numerical pipeline. The reader's weakest assumption identifies exactly this, and I agree. The paper's own text admits only that minimizers of the relaxation give a lower bound and that a coupled fixed point would solve the original; it offers no convergence or gap analysis. Since Algorithm 1's output is used for every reported comparison, a positive gap would make the empirical claims unsubstantiated. Thus I recommend no change to the reader's CONDITIONAL verdict.","tokens_in":21238,"tokens_out":7941,"duration_ms":77223,"concrete_test":"Use a tiny discrete instance (e.g., |X1|=|X2|=3, |Z|=4, random distance matrices and uniform weights). Compute the exact quadratic minimum F_lambda^* := min_alpha F_lambda(alpha,alpha) by enumerating all extreme points of the fixed-marginal transport polytope or by a global solver. Run Algorithm 1 from at least 10 random initializations, with the paper's epsilon=0.001 and several lambda values, until its stopping rule. Report (i) the relative gap (F_lambda(alpha1,alpha2) - F_lambda^*)/F_lambda^*, (ii) total-variation distance ||alpha1 - alpha2||_TV at termination, and (iii) the original objective value at the output plan alpha1. If the gap or ||alpha1 - alpha2|| is bounded away from 0, the relaxation is not certified tight and the numerical results should be repositioned as heuristic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The computational claim rests on an unverified equivalence between the quadratic problem (9) and the bilinear relaxation (10) as actually solved. Proposition 3.7 guarantees only that an exact minimizer of (9) gives projections solving (3), and that any solution of (3) can be lifted; it says nothing about the iterates of Algorithm 1. The text observes that a bi-convex minimizer with alpha1 = alpha2 solves (9), but block-coordinate descent on a bilinear objective has no convergence guarantee to such a coupled point: it can stall at a saddle or cycle with alpha1 != alpha2. Each Sinkhorn step also solves an epsilon-regularized linear subproblem, introducing an additional unquantified perturbation. Therefore the values and comparisons 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 lower-bound problem. Because the paper's third contribution is precisely that the bi-convex relaxation enables computation, this gap is load-bearing, not stylistic.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21460,"tokens_out":6538,"duration_ms":60823,"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":[{"comment":"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.","section":"Section 4, Eq. (10) and Algorithm 1"},{"comment":"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.","section":"Section 5 and Section 6.1"}],"minor_comments":[{"comment":"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.","section":"Proposition 3.7"},{"comment":"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).","section":"Section 4, first paragraph"},{"comment":"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.","section":"Proof of Proposition 3.4"},{"comment":"There are typographical errors: 'we we show' and 'isometies' should read 'we show' and 'isometries'.","section":"Proof of Proposition 3.2"},{"comment":"There are small typos: 'JMSD' should be 'JMDS' in Section 5, and 'KKN-Acc' in Table 2 should be 'KNN-Acc'.","section":"Section 5 and Table 2"},{"comment":"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.","section":"Figure 12 caption"},{"comment":"The text refers to the 'Swiss role' in one place; this should be 'Swiss roll'.","section":"Figure 4 caption and text"}],"recommendation":"major_revision","confidential_remarks":"The theoretical core of the paper is defensible and the existence/lambda-limit results are solid. The main obstacle is the unanalyzed gap between the quadratic problem and the bi-convex relaxation used in the numerical solver; this affects every reported number in Section 6 and should be addressed before publication. I do not see grounds for rejection, since the gap is addressable with additional analysis or with honest reporting of the relaxation gap and the alpha1-alpha2 discrepancy. The paper's code availability is a positive factor."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The theoretical core of this paper is genuinely useful and appears sound. The authors introduce a new functional EW_lambda that jointly embeds two metric measure spaces into a fixed compact metric space, with Gromov–Wasserstein penalties on the marginals. They prove existence of minimizers, and they show that as lambda goes to infinity the minimizers converge to a minimizer of the embedded Wasserstein distance (when isometric embeddings exist) or to GW approximations otherwise; as lambda goes to zero, they recover fixed-support GW barycenters. These are new results, and the proofs in Appendix A look correct to me: they use standard compactness and lower-semicontinuity arguments, and the chain of inequalities in Proposition 3.4 is solid. The reformulation as a quadratic multi-marginal unbalanced OT problem and the connection to JMDS are also well done and will be useful to people working on invariant OT and dimensionality reduction.\n\nThe main soft spot is the computational claim. The paper relaxes the quadratic problem (9) to the bilinear lower-bound problem (10) and then runs block-coordinate descent with a multi-marginal Sinkhorn scheme. But there is no proof that the alternating iterates converge to a coupled solution alpha1 = alpha2, nor any bound on the gap between the lower bound and the true minimum. So the numerical values in Section 6 could be minima of a strictly looser problem, not of EW_lambda itself. This is a real weakness because the paper advertises the bi-convex relaxation as the way to compute the model. I don't think it invalidates the theory, but the numerical results should be presented more cautiously unless the authors can tighten the equivalence or at least report how far alpha1 and alpha2 are from each other in practice. The hyperparameter selection on a validation split is fine, but error bars are missing in Table 2, and the comparisons with JMDS/SCOT are on a limited set of examples, so the empirical claims are suggestive rather than definitive.\n\nWho should read this: anyone working on Gromov–Wasserstein distances, multi-marginal OT, or joint embedding of heterogeneous datasets. The theory is worth citing on its own. My recommendation: send it to peer review, with the expectation that the authors will need to address the relaxation gap in the numerical section—either by proving a convergence result for the alternating scheme, or by repositioning Algorithm 1 as a heuristic and discussing the gap explicitly. As it stands, the paper is a good theoretical contribution with an unverified computational appendix.","headline":"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.","tokens_in":22010,"tokens_out":1935,"would_cite":true,"duration_ms":19604,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["joint metric space embedding","unbalanced optimal transport","Gromov-Wasserstein distance","embedded Wasserstein distance","multi-marginal optimal transport","bi-convex relaxation","metric measure spaces"],"falsifier":"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.","tokens_in":51,"feed_emoji":"🔀","tokens_out":11029,"duration_ms":283049,"temperature":0.7,"pith_summary":"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.","feed_headline":"GW-penalized transport provably aligns two datasets in one metric space","feed_subtitle":"The relaxation converges to the exact embedded Wasserstein alignment as the penalty grows, with proofs and experiments.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines Sturm's Gromov–Wasserstein distance, the isometric-embedding construction that the embedded Wasserstein distance relaxes.","marker":"(Sturm, 2006)"},{"why":"Provides the computable Gromov–Wasserstein distance used as the marginal penalization in the proposed functional.","marker":"(Mémoli, 2011)"},{"why":"Introduces the embedded Wasserstein distance in Gaussian mixture spaces, which the paper names, targets, and generalizes to arbitrary metric spaces.","marker":"(Salmona et al., 2024)"},{"why":"Defines the Wasserstein Procrustes problem, the Euclidean special case of the embedded Wasserstein distance.","marker":"(Grave et al., 2019)"},{"why":"Proposes joint multidimensional scaling, the free-support counterpart whose fixed-support version the paper derives from the same objective.","marker":"(Chen et al., 2023)"},{"why":"Supplies the unbalanced multi-marginal Sinkhorn scheme used as the inner solver in Algorithm 1.","marker":"(Beier et al., 2022)"},{"why":"Establishes fixed-support Gromov–Wasserstein barycenters, the limit object recovered as the penalty parameter goes to zero.","marker":"(Beier et al., 2023)"},{"why":"Provides the lemma used to pass the Gromov–Wasserstein approximations to the limit in the convergence proofs.","marker":"(Beier & Beinert, 2025)"}],"fun_headline_variants":["GW-penalized OT embedding provably converges as penalty grows","Provable joint metric space embedding from GW-penalized OT","Unbalanced OT with GW penalty: exact embedding via proven convergence","GW-penalized transport: provable convergence to exact metric embedding","Joint embedding via unbalanced OT with proven GW penalty convergence"],"cache_read_input_tokens":24192,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["GW-penalized OT embedding provably converges as penalty grows","Provable joint metric space embedding from GW-penalized OT","Unbalanced OT with GW penalty: exact embedding via proven convergence","GW-penalized transport: provable convergence to exact metric embedding","Joint embedding via unbalanced OT with proven GW penalty convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001837,"raw_usage":{"total_tokens":7191,"prompt_tokens":881,"completion_tokens":6310,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":6225}},"tokens_in":497,"tokens_out":6310,"duration_ms":39987,"temperature":1.0,"reasoning_tokens":6225,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T12:29:38.521544+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On the geometry of metric measure spaces","cited_arxiv_id":null,"evidence_quote":"Defines Sturm's Gromov–Wasserstein distance, the isometric-embedding construction that the embedded Wasserstein distance relaxes."},{"cited_title":"Gromov -- Wasserstein -like distances in the Gaussian mixture models space","cited_arxiv_id":null,"evidence_quote":"Introduces the embedded Wasserstein distance in Gaussian mixture spaces, which the paper names, targets, and generalizes to arbitrary metric spaces."},{"cited_title":"Unsupervised alignment of embeddings with W asserstein procrustes","cited_arxiv_id":null,"evidence_quote":"Defines the Wasserstein Procrustes problem, the Euclidean special case of the embedded Wasserstein distance."},{"cited_title":"Unsupervised manifold alignment with joint multidimensional scaling","cited_arxiv_id":null,"evidence_quote":"Proposes joint multidimensional scaling, the free-support counterpart whose fixed-support version the paper derives from the same objective."},{"cited_title":"Unbalanced multi-marginal optimal transport","cited_arxiv_id":null,"evidence_quote":"Supplies the unbalanced multi-marginal Sinkhorn scheme used as the inner solver in Algorithm 1."},{"cited_title":"Multi-marginal G romov-- W asserstein transport and barycentres","cited_arxiv_id":null,"evidence_quote":"Establishes fixed-support Gromov–Wasserstein barycenters, the limit object recovered as the penalty parameter goes to zero."},{"cited_title":"and Beinert, R","cited_arxiv_id":null,"evidence_quote":"Provides the lemma used to pass the Gromov–Wasserstein approximations to the limit in the convergence proofs."}],"review_version":1}