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REVIEW 5 major objections 5 minor 17 references

A dimensionality reduction technique based on the Gromov-Wasserstein distance

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A clean but incremental GW-based DR method whose headline empirical claim is undercut by the metric it chose to report. read the letter →

arxiv 2501.13732 v2 pith:3FPJ3GA2 submitted 2025-01-23 stat.ML cs.LG

Rafael P. Eufrazio , Eduardo Fernandes Montesuma , Charles C. Cavalcante This is my paper · ORCID
classification stat.MLcs.LG MSC 49Q2262H25
keywords dimensionalityreductionGromov-WassersteindistanceoptimaltransportmultidimensionalscalingIsomapgeodesicmanifoldlearninggradientdescent
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

5 major / 5 minor

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.

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 (5)
  1. [Section IV, Tables I and II] 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.
  2. [Section IV, Table I] 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.
  3. [Section IV, Tables I and II] 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.
  4. [Section II.B] 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.
  5. [Section III] 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.
minor comments (5)
  1. [Figure 3 and Section IV] 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.
  2. [Section IV, Table II] 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.
  3. [Algorithm 1] 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.
  4. [Section II.B] 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.
  5. [Figure 1] The abbreviation EWCA appears in Figure 1 before its definition in Section IV; define it at first use.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: GW-MDS optimizes a Gromov-Wasserstein objective that is not equal to the reported Pearson-correlation evaluation, and no load-bearing derivation reduces to its own inputs.

full rationale

The core derivation is Eq. (6), Y* = argmin_Y GW(mu, nu), a well-defined optimization problem whose objective, Eq. (5), is a transport-plan-weighted quadratic comparison of all distance pairs. The reported evidence, Tables I and II, uses the Pearson correlation coefficient, Eq. (7), which is invariant under monotone distance transformations. A minimizer of GW need not maximize Pearson, and high Pearson can coexist with poor GW loss, so the evaluation metric is not the optimized objective by construction. The paper also defines the MDS stress in Eq. (1) but never reports it; this is a missing-support problem for the claim that pairwise distances are better preserved, not a circularity. The alternating minimization strategy is justified by citing Afriat's classical 1971 Lagrange-theory paper, which is an independent external source, not a chain of self-citations. The one self-citation by a coauthor, Montesuma et al. [8], appears only as a general review pointer on optimal transport for machine learning and is not load-bearing for any stated result. The claim that the GW transport plan is sparse is an unverified assumption, but it does not identify any predicted quantity with a fitted input or define one variable in terms of another. Overall, the paper's central claim rests on empirical comparisons against MDS, Isomap, PCA, and EWCA, and those comparisons are not forced by the objective or by the citations. The correct criticism is that the decisive stress-based comparison is omitted, not that the derivation is circular.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The method introduces no new physical or mathematical entities. Its load-bearing assumptions are the suitability of GW as a DR objective, the unproven sparsity of the transport plan, and the convergence of the alternating minimization. The free parameters are the learning rate, iteration count, neighbor count for the geodesic variant, and initialization choice.

free parameters (4)
  • Learning rate eta = 0.1 and 0.01
    The paper tests two learning rates and reports that both converge to similar loss ranges; the choice affects convergence speed. This is a hyperparameter fitted by inspection of loss curves.
  • Number of gradient descent iterations = 600 (used in figures)
    Figures 6 and 7 show 600 iterations, but the stopping criterion is not stated; the number of iterations is a free choice per dataset.
  • Number of neighbors k for geodesic variant = Not specified
    The geodesic variant in Section IV requires constructing a nearest-neighbor graph, but the value of k is never given. This parameter directly affects the geodesic distance matrix and the resulting embedding.
  • Initialization strategy (randn or PCA) = Both tested
    The initialization choice affects convergence speed and the final embedding. The paper reports both are stable but does not specify which was used for the reported Tables.
assumptions (3)
  • domain assumption Minimizing the GW distance between the empirical measure of the data and the empirical measure of the embedding yields a useful low-dimensional representation.
    The whole method rests on this premise. No theorem or formal argument is given that solutions of Eq. (6) preserve the structure of the data in a way that is useful for DR. This is an assumption about the suitability of the GW objective.
  • ad hoc to paper The optimal transport plan is sparse, so GW 'boils down to a handful of non-zero elements.'
    Section II.B makes this claim without proof. In general, exact GW plans between uniform measures are not sparse; sparsity is a property of entropically regularized OT, not of the exact formulation used here. This assumption is used to argue that GW captures local relationships, but it is not justified.
  • standard math Gradient descent on the GW objective converges to a good local optimum.
    The algorithm uses gradient descent with automatic differentiation, but the GW objective is nonconvex and the alternating minimization has no convergence guarantee. The paper cites [11] as a general Lagrange reference without a derivation. This is an unproven background assumption about the optimization landscape.

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Cite this review

Pith. "Pith review of A dimensionality reduction technique based on the Gromov-Wasserstein distance." pith.science (2026). https://pith.science/paper/3FPJ3GA2

@misc{pith2026250113732,
  author       = {Pith},
  title        = {Pith review of: A dimensionality reduction technique based on the Gromov-Wasserstein distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3FPJ3GA2}},
  note         = {Machine review of arXiv:2501.13732}
}
read the original abstract

Analyzing relationships between objects is a pivotal problem within data science. In this context, Dimensionality reduction (DR) techniques are employed to generate smaller and more manageable data representations. This paper proposes a new method for dimensionality reduction, based on optimal transportation theory and the Gromov-Wasserstein distance. We offer a new probabilistic view of the classical Multidimensional Scaling (MDS) algorithm and the nonlinear dimensionality reduction algorithm, Isomap (Isometric Mapping or Isometric Feature Mapping) that extends the classical MDS, in which we use the Gromov-Wasserstein distance between the probability measure of high-dimensional data, and its low-dimensional representation. Through gradient descent, our method embeds high-dimensional data into a lower-dimensional space, providing a robust and efficient solution for analyzing complex high-dimensional datasets.

Figures

Figures reproduced from arXiv: 2501.13732 by the authors.

Figure 1
Figure 1. Qualitative analysis of dimensionality reduction algorithms. Better seen on screen. While PCA and Entropic Wasserstein [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Panels (a) and (d) show the toy manifolds, panels (b) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Scatter plot of pairwise distances in X and Y. Distances are computed between points in the faces dataset of [9]. Overall, GW-MDS yields embeddings that better preserve the pairwise distances in X . distances in the embedding space, dY , that is, ρ = cov(dX , dY ) σ(dX )σ(dY ) , (7) where cov(X, Y ) and σ(X) is the covariance and standard deviation for random variables X and Y . We summarize our quantitative analysi… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Panels (a) and (d) show the toy manifolds, panels (b) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 7
Figure 7. Figure 7: Comparison of the loss curves using two different learn [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 6
Figure 6. Figure 6: Comparison of loss curves under different initialization [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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Reference graph

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