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Fused Gromov-Wasserstein Variance Decomposition with Linear Optimal Transport

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper establishes a Pythagorean-style decomposition of Fréchet variance in 2-Wasserstein, Gromov-Wasserstein, and fused Gromov-Wasserstein spaces that makes the percentage of variance explained by linear optimal transport embeddings…

desk verdict A useful but modest LOT variance-decomposition extension, with a wrong displayed formula in Section 3.4 and a false isometry claim in the DTMRI section that need fixing before the numbers can be trusted. read the letter →

arxiv 2411.10204 v1 pith:IPNTQ2BH submitted 2024-11-15 stat.ME cs.LG

Michael Wilson , Tom Needham , Anuj Srivastava This is my paper · ORCID
classification stat.MEcs.LG MSC 62R2049Q22
keywords LinearOptimalTransportFréchetvarianceWassersteinbarycenterGromov-WassersteindistanceFuseddecompositionbarycentricprojectiondimensionalityreduction
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

The paper asks how much statistical information survives when probability measures are embedded into Euclidean space by Linear Optimal Transport (LOT), and answers by decomposing the Fréchet variance of a dataset of measures into a deterministic part carried by the LOT embedding and a residual probabilistic part. For the 2-Wasserstein distance, the decomposition is an exact identity: the squared distance splits into a squared distance to the barycentric projection plus a weighted residual, so the fraction of variance explained by the embedding becomes a computable number. The same split is proved for Gromov-Wasserstein distances, with the residual equal to a difference of squared 2-diameters, and is then extended to the fused Gromov-Wasserstein setting by linearity. The paper demonstrates on MNIST, IMDB-50000, and diffusion-tensor MRI data that low-dimensional LOT embeddings capture a large share of Fréchet variance and support competitive classifiers, giving practitioners a principled way to choose embedding dimension and the fusion parameter α.

What carries the argument

The load-bearing object is the barycentric projection $T(x_i)=\sum_j (\gamma_{ij}/a_i)y_j$, the conditional expectation of the target support point under the optimal coupling. It splits any optimal coupling into a deterministic part (from $\nu$ to $T_\#\nu$) and a purely probabilistic part (from $T_\#\nu$ to $\mu$) whose cross term vanishes by construction; in the Gromov-Wasserstein case the analogous projection $\mathbb{T}=(\mathbb{X},\omega_C,\nu)$ makes the identity coupling optimal and leaves the squared-diameter difference as the residual. This machinery converts a nonlinear variance question in Wasserstein space into a Euclidean variance question on the LOT embedding.

What would settle it

Take a small dataset of empirical measures on R whose true Wasserstein barycenter is known analytically, compute the deterministic component using a deliberately suboptimal coupling, and check whether it still equals $W_2^2(\nu, T_\#\nu)$ as claimed; more directly, compare the variance-explained percentage obtained with an exact barycenter against the percentage obtained with a free-support barycenter on the same data and measure the discrepancy.

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Extended reading notes

Core claim

On its own terms, the paper establishes that for an optimal coupling $\gamma$ and its barycentric projection $T$, the squared 2-Wasserstein distance splits exactly as $W_2^2(\nu,\mu)=W_2^2(\nu,T_\#\nu)+\sum_{ij}\gamma_{ij}\lVert T(x_i)-y_j\rVert^2$, and the squared 2-Gromov-Wasserstein distance splits as $GW_2^2(\mathbb{X},\mathbb{Y})=GW_2^2(\mathbb{X},\mathbb{T})+(\mathrm{diam}_2(\mathbb{Y})^2-\mathrm{diam}_2(\mathbb{T})^2)$. Averaging the first identity over a dataset makes the deterministic term exactly the sample Fréchet variance of the LOT-embedded measures, so the fraction of variance explained becomes a computable diagnostic. Corollary 1 extends the same split to the $\alpha$-fused 2-Gromov-Wasserstein distance, giving a variance-explained number for all three distances.

Load-bearing premise

The interpreted percentage-of-variance numbers assume the couplings used for barycentric projection are exactly optimal and that the free-support barycenter is a good approximation of the true Fréchet barycenter; with approximate algorithms (as in Definition 4 and the experiments of Sections 4.3–4.5) the deterministic term is only approximately a squared distance, so the reported percentages carry unquantified error.

Editorial extensions

If this is right

  • For any dataset of measures with an optimal coupling to a template, the ratio of the deterministic component to total Fréchet variance is an R-squared-style diagnostic for LOT embeddings, and it can be reported alongside any downstream classifier.
  • In the Gromov-Wasserstein case the residual has the closed form of a difference of squared 2-diameters, so embedding quality is summarised by two scalar quantities rather than a full coupling.
  • The F n,d statistic generalises one-way ANOVA to test equality of n-support barycentric projections, with permutation tests available when the null distribution is unknown.
  • The MNIST, IMDB-50000, and diffusion-tensor MRI experiments indicate that small support counts (7–100) can explain a large share of Fréchet variance while keeping classifier accuracy high, and the decomposition can guide the choice of n and α.

Reading between the lines

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

  • An extension the paper leaves implicit is that the deterministic fraction, as a function of n and α, could serve as an unsupervised model-selection criterion for embedding dimension before labels are used.
  • A natural robustness check that the paper does not run is to resample the data or rerun the barycenter algorithm many times and report a distribution of variance-explained values, turning the diagnostic into an interval estimate.
  • Because the exactness of the deterministic term depends on optimality of the coupling, applying the decomposition inside approximate or partial-transport pipelines should be preceded by an optimality check; the paper notes the issue for non-optimal couplings but does not quantify the resulting bias.
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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 / 5 minor

Summary. The paper develops a decomposition of the sample Fréchet variance of a set of empirical measures into a deterministic component (the variance captured by their barycentric projections onto a template) and a probabilistic residual. It proves the Wasserstein case, credited to earlier work, proves a Gromov-Wasserstein analogue, and extends the result to Fused Gromov-Wasserstein distances. The authors then use the fraction of variance in the deterministic component as a diagnostic for Linear Optimal Transport embeddings, and illustrate it on MNIST digit images, IMDB movie reviews, and Diffusion Tensor MRI tract data, together with a permutation-based extension of an ANOVA-type statistic.

Significance. The proposed R-squared-style diagnostic for LOT embeddings is useful, and the FGW variance decomposition appears novel; the paper correctly attributes the Wasserstein identity to prior work. The proofs in Appendices A.1 and A.2 are internally consistent, and the variance-explained ratio is derived from definitions rather than fitted, so there is no circularity in the central identity. The experiments are substantive and the DTMRI application is original. However, the manuscript currently contains a displayed decomposition that contradicts Corollary 1 and a non-isometric vectorization in the DTMRI section; both affect the reported numbers and must be fixed before the paper can be accepted.

major comments (2)
  1. [Section 3.4, displayed decomposition after Corollary 1] Corollary 1 gives FGW^2_{2,α}(X,Y^ℓ) = FGW^2_{2,α}(X,T^ℓ) + C^α_FGW(π^ℓ). Averaging over ℓ therefore yields dVar^n_{FGW_{2,α}} = (1/N)Σ_ℓ FGW^2_{2,α}(X,T^ℓ) + (1/N)Σ_ℓ C^α_FGW(π^ℓ). The displayed formula instead multiplies the two sums by α and 1−α. Since C^α_FGW already contains the α-weighting, the two expressions agree only when the omitted term is zero; for α=1/2 the displayed right-hand side is exactly half of dVar^n. The variance-explained percentages in Section 4 are ratios built from the deterministic term, so the authors must correct the displayed formula and recompute or confirm all affected numbers and figures.
  2. [Section 4.5.1, Eq. (10)] The mapping in Eq. (10) is not an isometry from (R^3 × Sym^+_3, product of the standard Euclidean metrics) to R^9. Under the Frobenius metric on symmetric matrices, off-diagonal entries should be scaled by sqrt(2), not by 2; with factor 2, the squared distance in R^9 adds 4Σ_{i<j}(Δm_ij)^2 instead of 2Σ_{i<j}(Δm_ij)^2. The LOT distances and DTMRI variance decomposition in Section 4.5 are therefore computed in a different metric than claimed. Please replace the factor 2 by sqrt(2), or explicitly define the nonstandard metric if that was intended.
minor comments (5)
  1. [Corollary 1 proof] The final equality in the proof of Corollary 1 uses [40, Lemma 1, part 1] without stating the lemma. Since this is the only step that replaces C^α_FGW(T) by the distance FGW^2(X,T), please state the needed lemma or give a proof so that the derivation is self-contained.
  2. [Section 4.3, first paragraph and Fig. 3 caption] The text says free-support Wasserstein barycenter algorithms [19,43] are used for all α, while Fig. 3 labels the templates as FGW barycenters. Please clarify which objective and algorithm produced the templates for each α, and avoid the term 'FGW barycenter' if the template does not actually minimize the FGW Fréchet variance.
  3. [Section 4.3 (Fig. 4) and Section 4.5 (Fig. 7)] The empirical curves are single runs without error bars or multiple initializations. Given the non-convex free-support barycenter procedure and the spike at n=8,9 noted in Section 4.3, report variability (for example, over initializations or bootstrap) for at least the main variance-explained and classification-accuracy curves.
  4. [Section 4.3 text and Fig. 4 caption] There is a cross-reference to the 'left panel of Figure 6' where Figure 4 appears to be meant, and the Fig. 4 caption says '2-Wasserstein variance' although the section analyzes FGW variance.
  5. [Theorem 1 proof and Eq. (7)] In the proof of Theorem 1, the line 'GW^2(ν, \tilde μ)' uses an undefined symbol \tilde μ; it should be GW^2(X,T). In Eq. (7), the index ℓ* in the numerator is undefined or a typo for ℓ.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity found: the decompositions are proved from definitions and external lemmas, and the reported 'variance explained' is a descriptive ratio rather than a fitted output.

full rationale

The derivation chain is self-contained. Proposition 1 is proved in Appendix A.1 from the optimal-coupling expansion and a transport-cost optimality argument; Theorem 1 is proved in Appendix A.2 by the analogous mechanism; Corollary 1 is a direct algebraic consequence of these two decompositions together with [40, Lemma 1], which is an external result, not authored by the present authors. The sample decomposition in Definition 4 and Eq. (6) is an averaged identity, and the 'percentage of variance explained' is the ratio of the deterministic term to the total Frechet variance: a defined descriptive statistic, not a parameter fitted to force any target. None of the experiments fit the decomposition to produce a pre-specified outcome; they solve optimal transport problems and then report the resulting ratio. The self-citations that appear ([16], [17], [54]) are only background application pointers in the introduction and are not load-bearing. The only substantive anomaly is the display in Section 3.4: dVar^n_FGW2,alpha := (1/N) Sum FGW^2(X,Y^ell) = alpha/N Sum FGW^2(X,T^ell) + (1-alpha)/N Sum C^alpha_FGW(pi^ell) is algebraically inconsistent with Corollary 1, which would give (1/N) Sum FGW^2(X,T^ell) + (1/N) Sum C^alpha_FGW(pi^ell). This is a correctness or typographical issue, not circularity, and it does not raise the circularity score.

Assumptions & free parameters 4 free parameters · 3 assumptions · 1 invented entities

The central theorem is derived with exact optimality assumptions. The experiments introduce user-chosen parameters n and α, a data-fitted λ*, uncontrolled initialization, and a novel CTDI representation, none of which are external benchmarks. This is a moderate number of assumptions for a methodological paper.

free parameters (4)
  • Support-point count n in free-support barycenter = varied between 1 and 200 depending on experiment
    Chosen by the user; determines LOT embedding dimension and strongly affects the variance decomposition and classification accuracy.
  • Fusion weight α = MNIST: 0, 0.25, 0.5, 0.75, 1; IMDB: 0, 5e-7, 2.5e-6, 1
    Chosen by hand to balance node-location and edge-structure components; IMDB values are selected to balance percent variance explained.
  • λ* in DTMRI appendix = tr((Z0)^T Z0) / (tr((Z1)^T Z1) + tr((Z0)^T Z0))
    Computed from the embedded data to balance location and covariance components; this is a data-fitted parameter.
  • Random initialization for free-support barycenter = not reported (standard normal points, log-uniform covariances)
    The free-support barycenter algorithm is non-convex; results depend on initialization, and no seed is given, so this is an uncontrolled choice.
assumptions (3)
  • domain assumption Optimal couplings and barycentric projections are computed exactly for the relevant distance (W2, GW, or FGW).
    The decomposition identities require γ to be an optimal coupling; in practice, linear programming and free-support algorithms return approximations. This is used in Definitions 4, 6, 8 and all experiments.
  • domain assumption The free-support barycenter algorithm of [19, 43] returns a valid Fréchet barycenter of the sample.
    The variance decomposition is computed with respect to this template; if the algorithm returns a poor local optimum, the percentage of variance explained is not the true Fréchet variance. See Section 2.2 and experiments.
  • domain assumption The Euclidean metric on Sym+_3 and the vectorization in Eq. (10) represent covariance distances faithfully.
    The DTMRI experiment relies on this for barycenter and LOT computations; the stated embedding is not isometric for the Frobenius metric, so this assumption is violated as written.
invented entities (1)
  • Covariance Tract Density Images (CTDI)
    purpose: Represent each white matter tract as an empirical measure on R3 × Sym+_3, coupling voxel locations with diffusion tensors, so that LOT can be applied.
    A new data representation introduced in Section 4.5.1; no external validation demonstrates that it preserves gender-relevant diffusion information better than existing representations.

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Cite this review

Pith. "Pith review of Fused Gromov-Wasserstein Variance Decomposition with Linear Optimal Transport." pith.science (2026). https://pith.science/paper/IPNTQ2BH

@misc{pith2026241110204,
  author       = {Pith},
  title        = {Pith review of: Fused Gromov-Wasserstein Variance Decomposition with Linear Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPNTQ2BH}},
  note         = {Machine review of arXiv:2411.10204}
}
read the original abstract

Wasserstein distances form a family of metrics on spaces of probability measures that have recently seen many applications. However, statistical analysis in these spaces is complex due to the nonlinearity of Wasserstein spaces. One potential solution to this problem is Linear Optimal Transport (LOT). This method allows one to find a Euclidean embedding, called LOT embedding, of measures in some Wasserstein spaces, but some information is lost in this embedding. So, to understand whether statistical analysis relying on LOT embeddings can make valid inferences about original data, it is helpful to quantify how well these embeddings describe that data. To answer this question, we present a decomposition of the Fr\'echet variance of a set of measures in the 2-Wasserstein space, which allows one to compute the percentage of variance explained by LOT embeddings of those measures. We then extend this decomposition to the Fused Gromov-Wasserstein setting. We also present several experiments that explore the relationship between the dimension of the LOT embedding, the percentage of variance explained by the embedding, and the classification accuracy of machine learning classifiers built on the embedded data. We use the MNIST handwritten digits dataset, IMDB-50000 dataset, and Diffusion Tensor MRI images for these experiments. Our results illustrate the effectiveness of low dimensional LOT embeddings in terms of the percentage of variance explained and the classification accuracy of models built on the embedded data.

Figures

Figures reproduced from arXiv: 2411.10204 by the authors.

Figure 1
Figure 1. Illustration of empirical measures µ1, µ2,.., µl and their barycenter ν. The measures T#ν l denote the barycen￾tric projections of ν towards µl . The vectors V l represent the Euclidean translations from ν to T#ν l 3 Fr´echet Variance Decompositions with LOT In this section, we present conditions under which Linear Optimal Transport can be used to decompose Fr´echet vari￾ance with respect to the Wasserstein, Gromov-… view at source ↗
Figure 2
Figure 2. Left: Sample data from 5 groups; Middle Left: p-values for permutations test for different values of n; Middle [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Top Row: Gaussian Kernel Reconstructions and graph representations for free support Fused Gromov [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Left: Components of decomposition calculated with respect to a free support Barycenter with different numbers [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Left: Components of decomposition calculated with respect to a free support Barycenter with different numbers [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Points represent locations in R 3 , (r, g, b) values correspond to (normalized) diagonal elements of covariance matrices. Left: free support barycenter of ’Fornix L’ in HCP data (200 support points); Middle: Subject 100307’s ’Fornix L’ in the HCP data (6980 support poi…
Figure 7
Figure 7. Figure 7: Left: Fr´echet variance and components of decomposition as a function of number of support points in free [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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