{"id":"078d1101-8c52-4eb7-977b-292f3189f508","arxiv_id":"2411.10204","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Squared Wasserstein, Gromov-Wasserstein, and Fused Gromov-Wasserstein distances decompose into a deterministic linear-optimal-transport component plus a residual, enabling a percentage-of-variance-explained diagnostic for LOT embeddings.","lead":"This paper derives a way to split the spread of a dataset of probability measures or graphs into a part captured by a linear optimal transport embedding and a leftover part, and demonstrates the split on MNIST, movie reviews, and brain MRI data. The split gives practitioners a variance-explained number they can use to choose how many support points to keep before applying classifiers to embedded data.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The FGW variance decomposition formula displayed in Section 3.4 is internally inconsistent with Corollary 1: it multiplies the deterministic and probabilistic terms by α and (1−α), so the two terms no longer sum to the total Fréchet variance.","rationale":"We focused on the internal inconsistency in the FGW variance decomposition because it directly undermines the paper's central contribution as stated. The mathematical core (Proposition 1, Theorem 1, Corollary 1) appears correct: the proofs are coherent, and the Pythagorean-type decomposition into deterministic and probabilistic components is a legitimate extension of known LOT and GW results. The weakness is in the presentation and application: the displayed FGW variance formula in Section 3.4 contradicts Corollary 1, so any reader who implements the displayed formula will obtain incorrect variance-explained percentages. This is a concrete, verifiable error, not an interpretational ambiguity. We also note the secondary issues raised by the reader: the DTMRI isometry claim in Eq. (10) is incorrect (off-diagonal matrix entries should be scaled by √2, not 2, to preserve the Frobenius norm), and the experiments lack error bars and code. These should be corrected, but the formula error is the most load-bearing because it is part of the central claim's statement. The approximation issue flagged in the reader's weakest_assumption is a genuine limitation—the free-support barycenter and approximate couplings introduce unquantified error in the empirical percentages—but it does not invalidate the exact decomposition for optimal couplings. Thus the verdict remains CONDITIONAL: the paper can be accepted after fixing the formula and the isometry, and ideally adding uncertainty quantification.","tokens_in":22036,"tokens_out":22221,"duration_ms":188979,"concrete_test":"Re-derive the displayed equation in Section 3.4 directly from Corollary 1 and Definition 9. Substitute α=0.5 into both sides for a synthetic pair (X,Y) where FGW^2(X,T) and C^α_FGW(π) are both nonzero; the displayed formula gives dVar = 0.5·(FGW^2(X,T)+C^α_FGW(π)) = 0.5·FGW^2(X,Y), contradicting Corollary 1. Alternatively, sum the two displayed terms and compare to (1/N)Σ FGW^2(X,Y^ℓ); they differ unless α=1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.4 displays the α-Fused Gromov-Wasserstein variance decomposition as dVar^FGW = (α/N)Σ FGW^2(X,T^ℓ) + ((1−α)/N)Σ C^α_FGW(π^ℓ). But Corollary 1 states FGW^2(X,Y^ℓ) = FGW^2(X,T^ℓ) + C^α_FGW(π^ℓ) for each ℓ, where C^α_FGW already includes the α weighting (Definition 9). Summing Corollary 1 and dividing by N gives dVar^FGW = (1/N)ΣFGW^2(X,T^ℓ) + (1/N)ΣC^α_FGW(π^ℓ). The displayed formula instead weights the two sums by α and (1−α), so the equality fails for any α ≠ 1 unless both sums are zero. For α=0.5, the RHS equals half the total variance, so the claimed 'decomposition' does not sum to dVar^FGW. This is not a typographical nuance: the percentages of variance explained reported in Section 4 (e.g., the elbow plots in Figs. 4 and 5) are computed as the ratio of the first term to the total; the missing 1/α factor would systematically undervalue the deterministic component for α<1. The correct formula follows immediately from Corollary 1 and should replace the displayed expression.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22368,"tokens_out":21799,"duration_ms":205297,"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":[{"comment":"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.","section":"Section 3.4, displayed decomposition after Corollary 1"},{"comment":"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.","section":"Section 4.5.1, Eq. (10)"}],"minor_comments":[{"comment":"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.","section":"Corollary 1 proof"},{"comment":"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.","section":"Section 4.3, first paragraph and Fig. 3 caption"},{"comment":"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.","section":"Section 4.3 (Fig. 4) and Section 4.5 (Fig. 7)"},{"comment":"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.","section":"Section 4.3 text and Fig. 4 caption"},{"comment":"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.","section":"Theorem 1 proof and Eq. (7)"}],"recommendation":"major_revision","confidential_remarks":"The two technical errors identified above are fixable, but the first directly affects the displayed variance decomposition and the second affects the DTMRI experiments; I do not see grounds for rejection. A reproducibility statement with code or data availability would strengthen the resubmission. The paper's fit with the journal's scope is reasonable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my take. The paper is a modest but useful extension of known LOT variance-decomposition results. Proposition 1 is correctly credited to [2,6]; the genuinely new part is carrying the decomposition into the GW and FGW settings and reading the deterministic term as an R-squared-style variance-explained measure for LOT embeddings. The proofs in Appendices A.1 and A.2 are internally consistent, and the permutation test around the F-statistic in Section 4.1 is a reasonable practical addition. The experiments span MNIST, IMDB, and DTMRI, and the qualitative conclusion—low-support LOT embeddings capture most of the variance—is plausible.\n\nThat said, there are three real problems.\n\nFirst, the displayed FGW variance decomposition in Section 3.4 is wrong as written. Corollary 1 gives FGW^2(X,Y) = FGW^2(X,T) + C^alpha_FGW(pi). Averaging and dividing by N yields (1/N) sum FGW^2(X,T) + (1/N) sum C^alpha_FGW(pi). The paper prints (alpha/N) sum FGW^2(X,T) + ((1-alpha)/N) sum C^alpha_FGW(pi), which fails to equal the total for alpha != 1. This isn't cosmetic; the elbow plots in Figures 4 and 5 are ratios built from these terms, so unless the code used the correct formula, the reported percentages are off by a factor of alpha. The paper needs to correct the formula and show the code or re-run the figures.\n\nSecond, the DTMRI embedding is not isometric. Mapping (10) uses 2v5, 2v6, 2v8; a Frobenius isometry would use sqrt(2). With 2, the product metric on R^3 x Sym+^3 is distorted, so the variance-explained numbers for DTMRI are computed under a different metric than claimed. Easy fix, but it changes the numbers.\n\nThird, the experimental reporting is thin. Point estimates come without error bars, no code is provided, and the free-support barycenter algorithm is approximate. That is a standard caveat, but it should be stated where the numbers live, not implied.\n\nThe formula and isometry issues are fixable, and the qualitative conclusions probably survive. But the paper as submitted cannot be taken at face value. It deserves a serious referee—the math core is sound and the application is useful—but I would send it back for major revision rather than desk reject.\n\nSincerely,","headline":"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.","tokens_in":22867,"tokens_out":6729,"would_cite":false,"duration_ms":58614,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62R20","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Linear Optimal Transport","Fréchet variance","Wasserstein barycenter","Gromov-Wasserstein distance","Fused Gromov-Wasserstein distance","variance decomposition","barycentric projection","dimensionality reduction"],"falsifier":"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.","tokens_in":21843,"feed_emoji":"📊","tokens_out":10527,"duration_ms":94915,"temperature":0.7,"pith_summary":"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.","feed_headline":"Percentage of variance explained by LOT embeddings is now computable","feed_subtitle":"A decomposition splits Wasserstein Fréchet variance into deterministic and residual parts, then extends to Gromov-Wasserstein and fused…","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original squared-Wasserstein decomposition that Proposition 1 restates and reproves.","marker":"[2]"},{"why":"Provides the proof technique (Lemma 2.1) used to show the deterministic part equals the squared distance.","marker":"[6]"},{"why":"Defines the Gromov-Wasserstein distance between networks and proves existence of optimal couplings used in Theorem 1.","marker":"[15]"},{"why":"Provides the free-support Wasserstein barycenter algorithm used in the experiments.","marker":"[19]"},{"why":"Defines barycentric projection in the fused Gromov-Wasserstein setting and supplies Lemma 1 used in Corollary 1.","marker":"[40]"},{"why":"Gives the Gromov-Wasserstein free-support barycenter algorithm that underlies the GW and FGW experiments.","marker":"[43]"},{"why":"Introduces Linear Optimal Transport embeddings and the barycentric projection pipeline this decomposition interprets.","marker":"[53]"}],"fun_headline_variants":["Exact variance split for LOT and fused Gromov-Wasserstein","Compute variance explained by LOT embeddings exactly","LOT embeddings' variance explained: now a computable number","New decomposition yields variance explained for LOT embeddings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact variance split for LOT and fused Gromov-Wasserstein","Compute variance explained by LOT embeddings exactly","LOT embeddings' variance explained: now a computable number","New decomposition yields variance explained for LOT embeddings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000194,"raw_usage":{"total_tokens":1386,"prompt_tokens":1011,"completion_tokens":375,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":311}},"tokens_in":627,"tokens_out":375,"duration_ms":4169,"temperature":1.0,"reasoning_tokens":311,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:51:57.864258+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability","cited_arxiv_id":null,"evidence_quote":"Supplies the original squared-Wasserstein decomposition that Proposition 1 restates and reproves."},{"cited_title":"Linear optimal par- tial transport embedding","cited_arxiv_id":null,"evidence_quote":"Provides the proof technique (Lemma 2.1) used to show the deterministic part equals the squared distance."},{"cited_title":"The Gromov– Wasserstein distance between networks and stable net- work invariants","cited_arxiv_id":null,"evidence_quote":"Defines the Gromov-Wasserstein distance between networks and proves existence of optimal couplings used in Theorem 1."},{"cited_title":"Fast computation of Wasserstein barycenters","cited_arxiv_id":null,"evidence_quote":"Provides the free-support Wasserstein barycenter algorithm used in the experiments."},{"cited_title":"On a linear fused Gromov-Wasserstein distance for graph struc- tured data","cited_arxiv_id":null,"evidence_quote":"Defines barycentric projection in the fused Gromov-Wasserstein setting and supplies Lemma 1 used in Corollary 1."},{"cited_title":"Gromov-Wasserstein averaging of kernel and distance matrices","cited_arxiv_id":null,"evidence_quote":"Gives the Gromov-Wasserstein free-support barycenter algorithm that underlies the GW and FGW experiments."},{"cited_title":"A linear optimal trans- portation framework for quantifying and visualizing variations in sets of images","cited_arxiv_id":null,"evidence_quote":"Introduces Linear Optimal Transport embeddings and the barycentric projection pipeline this decomposition interprets."}],"review_version":1}