{"id":"2015fc3f-4997-4676-8b30-7bbea351bf5d","arxiv_id":"2501.13732","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"GW-MDS reformulates multidimensional scaling as a Gromov-Wasserstein minimization problem and shows embeddings whose pairwise distances correlate with the original ones comparably to MDS and Isomap.","lead":"GW-MDS is a new dimensionality reduction algorithm that minimizes the Gromov-Wasserstein distance between high-dimensional data and its low-dimensional embedding. It reinterprets MDS and Isomap probabilistically and reports competitive distance preservation on several benchmarks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tables I and II report only Pearson correlation, never the MDS stress the paper itself defines as the distance-preservation criterion; since Pearson is invariant to monotone distance distortions, the central claim that GW-MDS preserves pairwise distances better than MDS/Isomap is unsubstantiated.","rationale":"I read the paper in good faith: it proposes a concrete, implementable algorithm that alternates between solving a Gromov-Wasserstein transport problem and gradient-descent updates on the embedding, with a geodesic variant for manifold data. The central claim is empirical: GW-MDS yields embeddings whose pairwise distances correlate with original distances better than MDS and Isomap. The reader's conditional verdict already identifies missing support (no code, no error bars, absent baselines, deferred theory). My stress-test focuses on a more specific and, I think, more decisive gap: the paper defines stress as a distance-preservation metric but never reports it, while relying on Pearson correlation, which does not measure distance preservation. This is not an internal inconsistency—the algorithm could still work—but it means the reported numbers do not establish the stated claim. A simple recomputation of stress would settle the issue. I therefore agree with the conditional verdict and do not recommend changing it. The secondary concerns raised by the reader (sparsity assumption, deferred theory) are real but less directly tied to the quantitative support; my concern complements rather than replaces theirs.","tokens_in":9194,"tokens_out":5719,"duration_ms":55842,"concrete_test":"Recompute the embeddings from Tables I and II and compute the normalized stress S = sum_{i<j}(dX_ij - dY_ij)^2 / sum_{i<j} dX_ij^2 for MDS, Isomap, PCA, EWCA, and GW-MDS on MNIST, Faces, Swiss Roll, S-Curve, Torus, Mobius, and Sphere. Report S alongside the Pearson rho. If GW-MDS's stress is not lower than MDS's on the majority of datasets, the claim 'better preserve pairwise distances' fails; if it is lower, the current evidence is at least consistent with the claim but still needs error bars and code.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central empirical claim is that GW-MDS embeddings better preserve pairwise distances than MDS and Isomap. The paper itself defines the stress in Eq. (1) as the distance-preservation criterion and states: 'We do so through two metrics, namely, the stress introduced in equation (1), and the Pearson correlation coefficient.' However, Tables I and II report only Pearson correlation coefficients; the stress is never reported. Pearson correlation is invariant under any strictly monotone transformation of distances, so it does not measure whether pairwise distances are actually preserved. An embedding whose distances are a highly distorted but monotone function of the original distances can have high Pearson correlation while having poor stress. Because MDS directly minimizes stress, a comparison in terms of stress is the decisive test of whether GW-MDS improves distance preservation; omitting it leaves the headline claim unsupported. This is a missing-support concern, not a mathematical contradiction, but it is load-bearing for the paper's central contribution.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes GW-MDS, a dimensionality reduction algorithm that minimizes the Gromov-Wasserstein distance between the empirical measure of the original high-dimensional data and the empirical measure of a low-dimensional embedding, using alternating optimization of the transport plan and gradient descent on the embedding. It also introduces a geodesic variant analogous to Isomap. The manuscript claims that these embeddings preserve pairwise distances better than MDS and Isomap, supported by Pearson correlation coefficients on toy manifolds, MNIST, and Faces.","tokens_in":9362,"tokens_out":4159,"duration_ms":36523,"significance":"The core idea of casting MDS as a Gromov-Wasserstein problem is interesting and connects two active research lines. The algorithm is simple to describe, and the experiments cover standard benchmark datasets. The manuscript does not ship code or proofs, and the empirical evidence as presented is not yet sufficient to support the headline claim; with additional experiments and corrected theoretical statements the approach could be a useful contribution.","major_comments":[{"comment":"The text says distances are quantified through both the stress from Eq. (1) and Pearson correlation, but the tables report only Pearson correlations. Because Pearson correlation is invariant to any strictly monotone transformation of distances, high values do not establish pairwise-distance preservation, which is the paper's central claim. Since MDS and Isomap directly minimize stress-like objectives, reporting the stress defined in Eq. (1) is the decisive comparison; its omission leaves the main claim unsupported.","section":"Section IV, Tables I and II"},{"comment":"GW-MDS is not consistently better than MDS in Table I: on Mobius it is lower (0.947 vs 0.952) and on Torus it ties (0.993). The text in Section IV states that GW-MDS 'consistently outperforms' MDS, which is contradicted by the table. The conclusion should be reworded or further evidence provided.","section":"Section IV, Table I"},{"comment":"The tables report single correlation values without error bars, multiple runs, or significance tests, and no comparison is made with existing GW-based DR methods [16], [17] despite their being cited as related. Without these, the claimed advantage over the current state of the art is not established.","section":"Section IV, Tables I and II"},{"comment":"The statement that the transport plan matrix is sparse and 'boils down to a handful of non-zero elements' is not true in general for exact optimal transport with uniform empirical measures: the marginals only constrain row and column sums, and optimal GW plans can be dense. This claim is used to motivate a connection between GW and local stress, so it should be either proved under stated assumptions or removed.","section":"Section II.B"},{"comment":"The claim that the alternating minimization strategy is 'theoretically justified via [11]' is not supported. Reference [11] is a general paper on Lagrange multipliers and does not address convergence of alternating nonconvex GW minimization; the manuscript should either provide a self-contained convergence or stationarity statement with assumptions or soften the claim.","section":"Section III"}],"minor_comments":[{"comment":"Figure 3's caption says distances are computed on the faces dataset, while the text says 'in the context of MNIST'; one of these is wrong.","section":"Figure 3 and Section IV"},{"comment":"The text says the geodesic variant was tested on 'four distinct datasets,' but Table II lists seven datasets (MNIST, Faces, Swiss roll, S-Curve, Torus, Mobius, Sphere); the text and table should be reconciled.","section":"Section IV, Table II"},{"comment":"Line 8, 'Y_i^* ← n ∑ π*_ij Y*_j', is ambiguous because the same symbol Y* is used for the output of gradient descent and for the aligned representation; use a different symbol and define the summation index.","section":"Algorithm 1"},{"comment":"Equation (5) defines GW using π* from Eq. (4); since Eq. (4) is itself a minimization, writing GW(μ,ν) = ... with π* is acceptable only if π* is a global minimizer. A brief clarification would avoid confusion.","section":"Section II.B"},{"comment":"The abbreviation EWCA appears in Figure 1 before its definition in Section IV; define it at first use.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the novelty overlap with [16] and [17] is not addressed in the manuscript; the authors cite these very close GW-DR papers but do not compare against them. This is worth probing during revision. Also, the paper says code will be released upon acceptance; I would ask for an anonymized reproducibility artifact."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a plausible but incremental paper. The core idea — minimize GW between the empirical measure of the data and the empirical measure of a low-dimensional embedding — appears in the paper's own references [16] and [17]. The headline empirical claim, that GW-MDS preserves pairwise distances better than MDS and Isomap, is not actually supported by the reported experiments, because the paper only reports Pearson correlation, not the stress defined in Eq. (1). Pearson correlation is invariant under monotone distance distortions, so a high rho says nothing about distance preservation. The stress-test note lands.\n\nWhat the paper does well: the probabilistic re-framing of MDS/Isomap is clean, the alternating scheme (GW plan via POT, then gradient step) is simple and reproducible, and the geodesic variant is a sensible extension. The authors cite [16] and [17] rather than ignoring them, which helps a reader place the contribution. I believe the algorithm likely works as described.\n\nSoft spots, in order of importance:\n1. The missing stress results. The paper explicitly says it evaluates with stress and Pearson, but Tables I and II only show Pearson. MDS directly minimizes stress; comparing on stress is the decisive test. Without those numbers the main claim is unsupported.\n2. The sparsity statement in Section II.B: \"since the transport plan matrix is sparse... this boils down to a handful of non-zero elements\" is not generally true for exact GW with uniform measures. That weakens the local-structure motivation.\n3. The theoretical justification is a citation to [11] with no derivation; Afriat (1971) is about Lagrange multiplier theory, not an obvious proof of convergence for this alternating scheme. A real argument or a more relevant citation is needed.\n4. No comparison with [16] or [17], the most relevant GW-based DR baselines. EWCA is linear, so it's not a strong control.\n5. No error bars, no multiple runs, and on Mobius, Sphere, and Faces the method is worse or no better than baselines. The gains on MNIST and geodesic-MNIST are tiny.\n\nNone of this is a contradiction; the method could be useful. But the evaluation needs to be rebuilt around stress, with error bars and proper baselines, and the sparsity and convergence claims need correction. This is a reasonable submission for a workshop or a methods-oriented venue, not for a top journal as is.\n\nI would send it to peer review, because the idea is implementable and the weaknesses are fixable, but I'd expect major revision. The authors need to release code, report stress, include [16]/[17], and tone down the claims.","headline":"A clean but incremental GW-based DR method whose headline empirical claim is undercut by the metric it chose to report.","tokens_in":9927,"tokens_out":3236,"would_cite":false,"duration_ms":28134,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","62H25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Replacing the MDS stress with the Gromov-Wasserstein distance between empirical measures of data and embedding yields embeddings whose pairwise distances correlate with the original distances as well as or better than MDS and Isomap.","keywords":["dimensionality reduction","Gromov-Wasserstein distance","optimal transport","multidimensional scaling","Isomap","geodesic distance","manifold learning","gradient descent"],"falsifier":"Compute the exact Gromov-Wasserstein transport plan between two uniform empirical measures of moderate size (for instance, 100 points sampled from a sphere or Swiss roll paired with their GW-MDS embeddings) and count the non-zero entries of $\\pi^\\star$: if the plan is dense rather than sparse, the paper's explanation that the GW objective reduces to a handful of local MDS-like comparisons does not hold. A second check is to run GW-MDS from the MDS solution as initialization and compare the resulting distance correlations with those reported for random initialization, which would separate the contribution of the GW objective from the contribution of the starting point.","tokens_in":8948,"feed_emoji":"📉","tokens_out":20373,"duration_ms":148603,"temperature":0.7,"pith_summary":"GW-MDS (Gromov-Wasserstein multidimensional scaling) is a dimensionality reduction method that swaps the stress objective of classical MDS for the Gromov-Wasserstein distance between two probability measures: the empirical measure of the high-dimensional data and the empirical measure of the low-dimensional embedding. The Gromov-Wasserstein distance compares the internal distance matrices of the two point clouds through an optimal transport plan rather than through direct point-to-point matching, which the authors read as a probabilistic view of MDS in which local relationships are captured by the transport plan. The optimization alternates between solving for the transport plan with the embedding fixed and taking a gradient step on the embedding with the plan fixed, and the experiments report Pearson correlations between original and embedded pairwise distances that match or exceed those of MDS, Isomap, PCA and the entropic-Wasserstein baseline EWCA. A geodesic variant, which feeds shortest-path distances on a neighbour graph into the cost, competes with Isomap on curved manifolds such as the Swiss roll and the S-curve, with correlations up to 0.9993.","feed_headline":"Matches or beats MDS at preserving geometry","feed_subtitle":"A transport-plan view of multidimensional scaling preserves distances as well as or better than MDS and Isomap.","key_machinery":"The central object is the Gromov-Wasserstein distance between empirical measures, a metric from optimal transport that compares two probability distributions supported on possibly different metric spaces by aligning their internal distance matrices. The load-bearing identity is the formal parallel between this distance and the MDS stress: $GW(\\hat\\mu,\\hat\\nu)=\\sum_{i,j,k,\\ell}(d_X(x_i,x_j)-d_Y(y_k,y_\\ell))^2\\pi^\\star_{i,j}\\pi^\\star_{k,\\ell}$ versus $\\sigma=\\sum_{i<j}(d_X(x_i,x_j)-d_Y(y_i,y_j))^2$, with the transport plan $\\pi^\\star$ doing the work that the fixed index pairing does in the stress. The algorithm that carries the argument is alternating minimization: solve the quadratic transport problem for $\\pi$ with $Y$ fixed, then update $Y_{it+1}=Y_{it}-\\eta\\nabla_Y GW(\\hat\\mu,\\hat\\nu_{it})$ with $\\pi$ fixed, a strategy whose theoretical justification is deferred to a cited reference rather than derived in the paper; the final alignment map $Y^\\star_i\\leftarrow n\\sum_{j=1}^n\\pi^\\star_{ij}Y^\\star_j$ re-establishes the correspondence between original points and their embedded images.","core_discovery":"The paper's central claim is that the Gromov-Wasserstein distance, $GW(\\hat\\mu,\\hat\\nu)=\\sum_{i,j,k,\\ell}(d_X(x_i,x_j)-d_Y(y_k,y_\\ell))^2\\pi^\\star_{i,j}\\pi^\\star_{k,\\ell}$ evaluated at the optimal transport plan $\\pi^\\star$ coupling the empirical measures $\\hat\\mu=n^{-1}\\sum_i\\delta_{x_i}$ and $\\hat\\nu=n^{-1}\\sum_j\\delta_{y_j}$, is a workable objective for dimensionality reduction: minimizing it over the embedding coordinates $Y$ produces representations that preserve the geometry of the original data. The paper's justification is the formal resemblance between this objective and the MDS stress $\\sum_{i<j}(d_X(x_i,x_j)-d_Y(y_i,y_j))^2$; where the stress compares distances over ordered pairs $i<j$, the GW objective compares all quadruples $(i,j,k,\\ell)$, weighted by the transport plan, which the authors argue encodes local relationships between objects. The algorithm solves the nested problem by alternating an exact GW transport-plan computation with a gradient descent step $Y_{it+1}=Y_{it}-\\eta\\nabla_Y GW(\\hat\\mu,\\hat\\nu_{it})$, starting from a Gaussian or PCA initialization, and finishes with an alignment step $Y^\\star_i\\leftarrow n\\sum_j\\pi^\\star_{ij}Y^\\star_j$ that restores point-to-point correspondence between original points and their embedded images.","pith_inferences":["Editorial inference: the sparsity assumption on exact GW plans is the fragile link; for uniform empirical measures the optimal plan is generically not sparse, so a direct count of non-zero entries in $\\pi^\\star$ on real runs would either confirm the local-structure story or push the method toward an entropically regularized plan.","Editorial inference: because GW alignment is a correspondence between distance matrices rather than between points, the method is naturally a visualization and manifold-unfolding tool; using it where point identity matters would require checking that the posterior alignment step preserves rather than permutes identities.","Editorial inference: the formal parallel between stress and GW suggests a recipe the paper touches without systematizing — any dissimilarity matrix can be dropped into the $d_X$ slot of the GW cost to generate a new DR method, with the geodesic variant here being one instance; testing other dissimilarities such as diffusion distances or learned affinities is a direct follow-up experiment."],"forward_implications":["The transport-plan view gives a genuine alternative to the stress: on the reported benchmarks GW-MDS matches or beats MDS in distance correlation (MNIST 0.646 vs 0.643, Faces 0.904 vs 0.897, Sphere 0.826 vs 0.781), so minimizing $GW$ is at least as good a recipe for geometry-preserving embeddings as minimizing the stress.","The geodesic variant extends the same prescription to manifold learning: it outscores Isomap on MNIST (0.7887 vs 0.7697), the Swiss roll (0.9993 vs 0.9986) and the S-curve (0.9993 vs 0.9988) in distance correlation, and trails Isomap only on Faces and Sphere.","Because the objective aligns two distance matrices through a coupling rather than through a fixed labelling, the method outputs both an embedding and an explicit transport plan, and the final alignment step is what converts the plan back into a point-to-point map.","The method is insensitive to initialization in the sense that both random and PCA starts converge to stable embeddings, with the PCA start converging faster in the reported loss curves.","Each iteration costs an $O(n^3)$ transport-plan solve, so practical use is bounded to at most a few thousand points unless a parametric or minibatch variant is developed."],"supporting_citations":[{"why":"Supplies the MDS stress objective (equation 1) that the paper reinterprets and uses as the primary baseline.","marker":"[1]"},{"why":"Introduces the Gromov-Wasserstein distance (equations 4 and 5) that forms the paper's optimization objective.","marker":"[3]"},{"why":"Supplies the Isomap algorithm, the geodesic-distance construction, and the faces dataset used in the comparisons.","marker":"[9]"},{"why":"The source for the discrete optimal transport formulation, the discussion of sparse transport plans invoked in Section II.B, and the computational foundation of the algorithm.","marker":"[10]"},{"why":"Cited as the theoretical justification for the alternating minimization strategy, although the paper defers the derivation to this reference.","marker":"[11]"},{"why":"The Python Optimal Transport toolkit used to solve the GW transport-plan step inside Algorithm 1.","marker":"[13]"},{"why":"Defines the EWCA baseline that the experiments compare against alongside MDS and Isomap.","marker":"[14]"},{"why":"Supplies the MNIST benchmark, one of the two real-world datasets whose distance-correlation results carry the central comparison.","marker":"[15]"}],"fun_headline_variants":["Gromov-Wasserstein takes on MDS for data geometry","Transport-plan view of MDS preserves geometry better","New DR method beats MDS via optimal transport","Probabilistic MDS via Gromov-Wasserstein distance","GW distance: a fresh take on multidimensional scaling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method works only if minimizing the Gromov-Wasserstein distance between the empirical measures of data and embedding is a faithful stand-in for preserving pairwise distances, only if the alternating transport-plan and gradient scheme reaches a good embedding, and only if the transport plan between two uniform empirical measures is sparse enough for the GW objective to behave like a local MDS cost, a sparsity claim the paper asserts in Section II.B without proof.","fun_headline_variants_meta":{"raw":{"variants":["Gromov-Wasserstein takes on MDS for data geometry","Transport-plan view of MDS preserves geometry better","New DR method beats MDS via optimal transport","Probabilistic MDS via Gromov-Wasserstein distance","GW distance: a fresh take on multidimensional scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1553,"prompt_tokens":986,"completion_tokens":567,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":488}},"tokens_in":602,"tokens_out":567,"duration_ms":5302,"temperature":1.0,"reasoning_tokens":488,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:38:33.448053+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact Gromov-Wasserstein transport plan between two uniform empirical measures of moderate size (for instance, 100 points sampled from a sphere or Swiss roll paired with their GW-MDS embeddings) and count the non-zero entries of $\\pi^\\star$: if the plan is dense rather than sparse, the paper's explanation that the GW objective reduces to a handful of local MDS-like comparisons does not hold. A second check is to run GW-MDS from the MDS solution as initialization and compare the resulting distance correlations with those reported for random initialization, which would separate the contribution of the GW objective from the contribution of the starting point.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the MDS stress objective (equation 1) that the paper reinterprets and uses as the primary baseline."},{"cited_title":"Gromov–wasserstein distances and the metric ap- proach to object matching,","cited_arxiv_id":null,"evidence_quote":"Introduces the Gromov-Wasserstein distance (equations 4 and 5) that forms the paper's optimization objective."},{"cited_title":"A global geometric framework for nonlinear dimensionality reduction,","cited_arxiv_id":null,"evidence_quote":"Supplies the Isomap algorithm, the geodesic-distance construction, and the faces dataset used in the comparisons."},{"cited_title":"Computational optimal transport: With applications to data science,","cited_arxiv_id":null,"evidence_quote":"The source for the discrete optimal transport formulation, the discussion of sparse transport plans invoked in Section II.B, and the computational foundation of the algorithm."},{"cited_title":"Theory of maxima and the method of lagrange,","cited_arxiv_id":null,"evidence_quote":"Cited as the theoretical justification for the alternating minimization strategy, although the paper defers the derivation to this reference."},{"cited_title":"Pot: Python optimal transport,","cited_arxiv_id":null,"evidence_quote":"The Python Optimal Transport toolkit used to solve the GW transport-plan step inside Algorithm 1."},{"cited_title":"En- tropic wasserstein component analysis,","cited_arxiv_id":null,"evidence_quote":"Defines the EWCA baseline that the experiments compare against alongside MDS and Isomap."},{"cited_title":"The mnist database of handwritten digits,","cited_arxiv_id":null,"evidence_quote":"Supplies the MNIST benchmark, one of the two real-world datasets whose distance-correlation results carry the central comparison."}],"review_version":1}