{"id":"133147dd-108e-48c5-9fd6-92ddad3b0d3a","arxiv_id":"2507.13194","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper defines relation-aware slicing distributions and two new sliced Gromov-Wasserstein distances, RASGW and IWRASGW, with theoretical and empirical analysis.","lead":"This paper introduces a new way to choose the directions used when comparing two datasets from different spaces, called relation-aware slicing. The resulting distances are claimed to be faster and often more accurate than the standard sliced Gromov-Wasserstein distance in cross-domain alignment tasks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.8(iii) is false: IWRASGW's L→∞ limit is a σRA-weighted expectation, not EBSGW, because the estimator omits the proposal-density correction; a stated theoretical justification of IWRASGW is unsupported.","rationale":"I agree with the CONDITIONAL verdict but not with the reader's choice of weakest assumption. The sample-complexity step flagged by the reader is not the main weak point: the inequality Max-SGW_p ≤ c Max-SW_p follows from the elementary bound | |x−x′| − |y−y′| | ≤ |x−y| + |x′−y′| applied to one-dimensional projections, giving GW_p ≤ 2 W_p and a universal constant c=2; Proposition 4.9 can be repaired without changing the O(n^{−1/2}) rate. The genuinely load-bearing problem is Proposition 4.8(iii). The IWRASGW estimator defined in Definition 4.6 is a self-normalized importance average with proposal σRA and unnormalized weights f(g), so its L→∞ limit is the σRA-tilted ratio, not EBSGW. The proof in Appendix A.2(iii) jumps to the EBSGW expression without deriving the necessary density ratio. The two-point axis-aligned example makes the failure explicit: the relation-aware centers sit exactly on the zero set of g, so for large κ the IWRASGW limit is near zero while EBSGW is strictly positive. Since Proposition 4.8 is presented as the formal hierarchy that theoretically justifies IWRASGW's adaptiveness, a false statement in that proposition removes a key theoretical support. The empirical results may still be valid, and the correct limit can be stated and analyzed, but the paper as written needs a correction. This keeps the appropriate verdict at CONDITIONAL, so no adjustment to the reader's verdict is needed.","tokens_in":34055,"tokens_out":18541,"duration_ms":212935,"concrete_test":"Implement the two-point example µ = (δ_{e1}+δ_{−e1})/2, ν = (δ_{e2}+δ_{−e2})/2 in R^2 with f(x)=e^x. For θ=(cos φ, sin φ), compute g(θ)=GW_2^2(θ♯µ,θ♯ν) by solving the 2×2 GW problem; g is zero exactly when |cos φ|=|sin φ| and positive elsewhere, and the RAPD centers from Definition 4.3 are (±e1±e2)/√2, all zero directions. Then (a) compute EBSGW = E_{U(S^1)}[g e^g] / E_{U(S^1)}[e^g] by quadrature over φ; (b) for κ=200, estimate R_κ = E_{σRA}[g e^g] / E_{σRA}[e^g] by Monte Carlo sampling from the mixture of vMF or PS densities. R_κ is strictly smaller than EBSGW and tends to 0 as κ→∞, contradicting Proposition 4.8(iii). A short numerical script settles the disagreement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.8(iii) is false as stated. Definition 4.6 draws θ1,...,θL i.i.d. from σRA(·;µ,ν,σκ) and sets w_l = f(GW_p^p(θl♯µ, θl♯ν)); the normalized weighted sum therefore converges, as L→∞, to R(µ,ν) = E_{θ∼σRA}[g(θ) f(g(θ))] / E_{θ∼σRA}[f(g(θ))], where g(θ)=GW_p^p(θ♯µ,θ♯ν). EBSGW (Appendix B, Eq. 5) is E_{θ∼σ_μν}[g(θ)] with σ_μν(θ) ∝ f(g(θ)) with respect to the uniform law on S^{d−1}, i.e., E_U[g f(g)]/E_U[f(g)]. These two quantities agree only when σRA = U(S^{d−1}); for the vMF/PS location-scale construction with κ>0 and data-dependent centers, σRA is not uniform, and the missing importance ratio f(g)/σRA appears nowhere in Definition 4.6 or in the proof in Appendix A.2(iii). A concrete contradiction: take µ = (δ_{e1}+δ_{−e1})/2 and ν = (δ_{e2}+δ_{−e2})/2. The RAPD centers are (±e1±e2)/√2, exactly the directions where g(θ)=0, so as κ→∞ the L→∞ limit of IWRASGW tends to 0, while EBSGW = E_U[g e^g]/E_U[e^g] > 0. Thus the claimed convergence to EBSGW fails. The reader's sample-complexity concern is less serious: for 1D projections, | |x−x′| − |y−y′| | ≤ |x−y| + |x′−y′| gives GW_p ≤ 2 W_p, hence Max-SGW_p ≤ 2 Max-SW_p, so the inequality cited from Zhang et al. is repairable with constant 2.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces the relation-aware projecting direction (RAPD), a data-dependent distribution over the unit sphere constructed from pairwise differences of samples from the two input measures, and uses it to define two new sliced Gromov-Wasserstein distances, RASGW and IWRASGW. The main theoretical claims are semi-metricity (Theorem 4.7), a hierarchy relating RASGW/IWRASGW to SGW, Max-SGW, and EBSGW (Proposition 4.8), a sublinear sample complexity O(n^{-1/2}) (Proposition 4.9), and Monte Carlo error control (Proposition 4.10). The experimental section adapts several sliced Wasserstein methods to the GW setting and reports improved GW-2 distances in cross-domain generative tasks, as well as competitive FID/PSNR scores in autoencoder experiments.","tokens_in":34522,"tokens_out":10628,"duration_ms":112269,"significance":"The idea of an optimization-free, easily sampleable slicing distribution that depends on the geometry of both measures is a useful and timely contribution: it directly addresses the inefficiency of uniform slicing in high dimensions and opens a new family of sliced GW distances. The manuscript provides extensive experiments with several adapted baselines and makes code publicly available, which is a strength. However, the theoretical development is only partially sound: the claimed convergence of IWRASGW to EBSGW is incorrect as stated, and the sample complexity proof relies on an unproven inequality. These issues are load-bearing for the positioning of the proposed distances, but they appear repairable within the scope of a major revision.","major_comments":[{"comment":"The claim that IWRASGW converges to EBSGW as L→∞ is false for the stated definition. Definition 4.6 draws θ1,...,θL i.i.d. from σRA and uses self-normalized weights w_l = f(GW_p^p(θl♯µ, θl♯ν)). By the strong law, the limit is E_{θ∼σRA}[g(θ)f(g(θ))]/E_{θ∼σRA}[f(g(θ))], where g(θ)=GW_p^p(θ♯µ,θ♯ν). EBSGW, defined in Equation (5) of Appendix B, is E_{θ∼σ_{µ,ν}}[g(θ)] with σ_{µ,ν}(θ) ∝ f(g(θ)) relative to the uniform law on S^{d−1}, i.e., E_U[g f(g)]/E_U[f(g)]. These two quantities agree only when σRA equals the uniform law. A concrete counterexample is given by µ=(δ_{e1}+δ_{−e1})/2 and ν=(δ_{e2}+δ_{−e2})/2, for which the RAPD centers are (±e1±e2)/√2, where g(θ)=0. As κ→∞, the L→∞ limit of IWRASGW tends to 0, whereas EBSGW remains strictly positive. The proof in Appendix A.2(iii) simply asserts the convergence to EBSGW without addressing the missing proposal-density correction. This is a load-bearing claim: the abstract and Section 4.2 use it to justify IWRASGW as an energy-based slicing method.","section":"Proposition 4.8(iii) and Appendix A.2(iii)"},{"comment":"The sample complexity proof relies on the inequality Max-SGW_p(µ_n,µ) ≤ c Max-SW_p(µ_n,µ), attributed to Zhang et al. [2024] without stating the constant or the required conditions. While this inequality is repairable for 1D projections (since GW_p ≤ 2 W_p gives Max-SGW_p ≤ 2 Max-SW_p), the proof as written is not self-contained. Additionally, the step Max-SW_p^p(µ_n,µ) ≤ diam(X) max_{x,θ} |F_{n,θ}(x)−F_θ(x)|^p is not a standard bound and appears to need a more careful treatment of the diameter and the p-th root in the final rate. The claimed O(n^{-1/2}) rate may be correct, but the derivation must be rewritten with explicit constants and assumptions.","section":"Proposition 4.9 and Appendix A.3"},{"comment":"The quasi-triangle inequality is stated using objects RASGW_p^p(µ1,µ3;σκ,µ1,µ2) and RASGW_p^p(µ3,µ2;σκ,µ1,µ2) that are never defined in the main text. The notation suggests the slicing distribution is built from the pair (µ1,µ2) rather than from the measures being compared, but this is not stated explicitly. A precise definition of these quantities is needed for the statement to be meaningful.","section":"Theorem 4.7"},{"comment":"The claim that the Monte Carlo gradient estimator is unbiased requires a reparameterization derivation for the full sampling process, including the dependence of the RAPD centers on µ_ϕ through X,X′∼µ_ϕ. The text states 'with parameterized sampling' and then writes the estimator, but does not provide the reparameterization or a proof of unbiasedness. Since this estimator is used in the GWGAN training, a rigorous statement (or a citation to a full derivation) is needed.","section":"Section 4, Gradient Estimation"}],"minor_comments":[{"comment":"In the proof for IWRASGW, the denominator in the displayed expression is written as Σ_j GW_p^p(θj♯µ, θj♯ν) instead of Σ_j f(GW_p^p(θj♯µ, θj♯ν)); the correct denominator appears in Definition 4.6.","section":"Appendix A.1"},{"comment":"The formatting of Table 2 is garbled: the column labels for C and O are not cleanly aligned with the reported FID, PSNR, time, and epoch entries.","section":"Table 2"},{"comment":"The statement that the variance under σRA is 'easy to control' and 'we may always observe a tighter bound than SGW' is not justified; the variance of GW_p^p under σRA could be larger than under the uniform distribution for some measures, so the comparison should be phrased conditionally.","section":"Proposition 4.10"},{"comment":"The discussion of solving the 1D GW problem with identity or anti-identity permutations cites Vayer et al. [2019] and Beinert et al. [2023] and should clarify that this is a computational heuristic in practice, not a proven property of RASGW for arbitrary p and supports.","section":"Appendix D, Algorithm 5"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the incorrect limit claim in Proposition 4.8(iii). I recommend requiring either a corrected statement (the self-normalized importance sampling limit with proposal σRA) or a modification of Definition 4.6 to include the importance ratio 1/σRA, which would make IWRASGW a proper estimator of EBSGW but would require handling the normalizing constant. The sample complexity proof is repairable, and the experimental contribution is useful. The paper is within the scope of the journal, and the authors should be given the opportunity to fix these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: the paper has a genuinely new idea—relation-aware slicing for sliced Gromov-Wasserstein—and the empirical results are suggestive. But one of the three headline theoretical claims is false. Proposition 4.8(iii) states that IWRASGW converges to EBSGW as L→∞. It doesn't. The limit is a σRA-weighted expectation, E_{σRA}[g·f(g)]/E_{σRA}[f(g)], not the uniform-proposal EBSGW. The stress-test counterexample (two-point measures on opposite axes) is correct: as κ→∞ the IWRASGW limit goes to 0 while EBSGW stays positive. The proof in A.2(iii) simply asserts the convergence without the importance ratio. This is a real flaw, though not fatal to the whole paper.\n\nWhat's new: RAPD/RASD is a clean, optimization-free way to bias slices toward directions that preserve pairwise geometry. Extending RPSW to GW and building adapted SW baselines (DSGW, EBSGW, RPSGW, IWRPSGW) is useful, and the hierarchy in (i) and (ii) is fine: RASGW ≤ IWRASGW ≤ Max-SGW, and κ→0 recovers SGW. The MC error bound (Prop 4.10) is standard but correctly stated.\n\nSoft spots: the false limit is the big one. The sample-complexity proof also leans on Max-SGW ≤ c·Max-SW cited to Zhang et al. with no constant or conditions. That's repairable—a triangle-inequality argument gives the bound with c=2 (p-dependent constant)—so it's a presentation gap, not a fatal one. The 'quasi-triangle inequality' in Theorem 4.7 uses a σRA built on (μ1,μ2), which is not a true triangle inequality; calling it 'quasi' is fair but the metric status is weaker than marketed. Finally, the GWAE table lacks error bars, and the Omniglot PSNR claim in the text doesn't match the table (SGW is actually best there). Minor but sloppy.\n\nBottom line: the core construction is worth refereeing, and the false proposition can be fixed—IWRASGW remains a well-defined distance, just not a limit of EBSGW. A careful revision that corrects the limit statement and tightens the sample-complexity citation would make this citable. Send it to review, but make sure the theory referee knows to check the importance-weighting.\n\n— [Name]","headline":"A genuinely new slicing construction for sliced GW, but one of the three headline theoretical claims (Proposition 4.8(iii)) is wrong and needs correction before the paper can be accepted.","tokens_in":35053,"tokens_out":4712,"would_cite":false,"duration_ms":53812,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces relation-aware slicing distributions that replace uniform random directions in sliced Gromov-Wasserstein distances, proving semi-metric properties and square-root sample complexity while lowering GW-2 in cross-domain…","keywords":["relation-aware slicing","sliced Gromov-Wasserstein","Gromov-Wasserstein distance","cross-domain alignment","optimal transport","semi-metric","sample complexity","Monte Carlo estimation"],"falsifier":"Compute the ratio $\\mathrm{Max\\text{-}SGW}_2(\\mu_n,\\mu)/\\mathrm{Max\\text{-}SW}_2(\\mu_n,\\mu)$ on a compactly supported two-cluster Gaussian in $\\mathbb{R}^3$ as $n$ grows; Proposition 4.9 requires this ratio to stay bounded by a universal constant, so a ratio that grows with $n$ or with cluster separation would falsify the claimed statistical rate.","tokens_in":33852,"feed_emoji":"🎯","tokens_out":12508,"duration_ms":123350,"temperature":0.7,"pith_summary":"This paper argues that uniform random projection directions waste most of the computation in sliced Gromov-Wasserstein (SGW) distances, because most slices distort the pairwise-distance structure that cross-domain alignment is trying to match. It replaces uniform slices with a relation-aware slicing distribution built from pairs of samples from both domains, producing two new distances, RASGW and IWRASGW, that need no optimization to sample. The paper proves these remain semi-metrics with the same identity-of-isometry property as Gromov-Wasserstein, and that their empirical estimates converge at $O(n^{-1/2})$ under compact support. In cross-domain generative experiments, they report lower Gromov-Wasserstein-2 values than SGW, Max-SGW, DSGW, EBSGW, and RPSGW at comparable computational cost. A sympathetic reader would care because this is a parameter-free way to focus slicing on directions that preserve relational geometry, which is exactly what cross-domain alignment demands.","feed_headline":"Relation-aware slicing beats uniform directions for Gromov-Wasserstein","feed_subtitle":"Optimization-free slicing preserves metric and sample guarantees while improving cross-domain generative alignment.","key_machinery":"The load-bearing object is the Relation-Aware Projecting Direction (RAPD), a random unit vector obtained by taking two displacement vectors, one from each domain, normalizing them, adding and subtracting, and perturbing the normalized sum or difference by a von Mises-Fisher or power-spherical location-scale distribution. Its role is to concentrate the slicing distribution on directions where both pairwise distances shrink by the same cosine factor, so the projection does not systematically distort the relational discrepancy between the two spaces. RASD is just the marginal law of RAPD over data samples, which makes Monte Carlo sampling cheap: draw $X,X'$ from $\\mu$, $Y,Y'$ from $\\nu$, compute the normalized paths, toss a coin for sum or difference, then sample the perturbation.","core_discovery":"The central claim is that a data-dependent slicing distribution, rather than the uniform law on the sphere, should define sliced Gromov-Wasserstein distances. For independent pairs $(X,X')\\sim \\mu$ and $(Y,Y')\\sim \\nu$, the paper defines a Relation-Aware Projecting Direction as a location-scale perturbation around the normalized sum or difference of the displacement vectors $X-X'$ and $Y-Y'$; marginalizing over data gives the Relation-Aware Slicing Distribution (RASD). RASGW is the expectation of Gromov-Wasserstein between one-dimensional projections drawn from RASD, and IWRASGW reweights the sampled projections by an increasing function of their projected GW value. Theorem 4.7 establishes non-negativity, symmetry, identity of isometric isomorphism, and a quasi-triangle inequality; Proposition 4.8 places the new distances between SGW and Max-SGW, recovering SGW as $\\kappa \\to 0$ and approaching EBSGW as $L \\to \\infty$; Proposition 4.9 gives sample complexity $C\\sqrt{(d+1)\\log(n+1)/n}$. Experiments on GW-GAN and GW-autoencoder tasks support the claim that the relation-aware focus lowers the attained GW-2 distance without adding optimization overhead.","pith_inferences":["Editorial inference: RAPD could replace the uniform sphere as a proposal for importance-sampled energy-based slicing, potentially cutting the number of slices needed for a fixed approximation accuracy; the paper does not test this.","Editorial inference: since RAPD only requires paired displacement vectors, a relation-aware slice could be built on manifolds or graphs by substituting geodesic or shortest-path displacements for Euclidean differences; the paper proves its properties only for Euclidean projections.","Editorial inference: because max-slicing optimizes the worst direction rather than a relation-preserving one, the hierarchy in Proposition 4.8 suggests RAPD-based alignment will beat Max-SGW more clearly in higher dimensions, where uniform directions concentrate near orthogonal to any informative direction."],"forward_implications":["RASGW and IWRASGW serve as drop-in losses for GW-GANs and Gromov-Wasserstein autoencoders with the same time and memory complexity as SGW, $O(M n \\log n + M d n)$ and $O(M d + M n)$, and the reported GW-2 values are lower than all tested sliced-GW baselines.","As $\\kappa \\to 0$, RASGW converges to SGW; as $L \\to \\infty$, IWRASGW converges to EBSGW, so the new family interpolates across the existing sliced-GW spectrum.","Both distances are semi-metrics whose zero set is the class of isometrically isomorphic spaces, and they satisfy a quasi-triangle inequality, preserving the matching semantics of Gromov-Wasserstein.","The empirical estimator converges at the same $O(n^{-1/2})$ rate as conventional SGW under compact support, so focusing slices on informative directions costs no statistical efficiency in sample size.","The Monte Carlo error is bounded by the standard deviation of projected GW under RASD divided by $\\sqrt{M}$; varying $\\kappa$ gives a direct handle on estimator variance without extra computation."],"supporting_citations":[{"why":"Defines the Gromov-Wasserstein distance and its isometry-based metric properties that RASGW and IWRASGW aim to preserve.","marker":"[Mémoli, 2011]"},{"why":"Introduces sliced Gromov-Wasserstein via Radon projections and supplies the proof template for zero-distance implies isometric isomorphism.","marker":"[Vayer et al., 2019]"},{"why":"Provides the random-path projecting direction idea, the optimization-free slicing precedent, and the additive-constant fix for undefined normalized differences.","marker":"[Nguyen et al., 2024]"},{"why":"Establishes energy-based slicing and the rearrangement lemma used in Proposition 4.8 to bound RASGW by IWRASGW.","marker":"[Nguyen and Ho, 2023]"},{"why":"Supplies the inequality that bounds max-sliced Gromov-Wasserstein by a constant times max-sliced Wasserstein, underpinning the sample complexity proof of Proposition 4.9.","marker":"[Zhang et al., 2024]"},{"why":"Introduces the power spherical distribution and its differentiable sampling, used for reparameterized gradient estimation.","marker":"[De Cao and Aziz, 2020]"},{"why":"Provides the von Mises-Fisher distribution used as the location-scale perturbation around RAPD directions.","marker":"[Jupp and Mardia, 1979]"},{"why":"Defines the Gromov-Wasserstein GAN testbed in which RASGW and IWRASGW are evaluated.","marker":"[Bunne et al., 2019]"},{"why":"Defines the Gromov-Wasserstein autoencoder framework used for image reconstruction experiments.","marker":"[Nakagawa et al., 2023]"},{"why":"Establishes the square-root sample complexity of conventional sliced Wasserstein that Proposition 4.9 matches.","marker":"[Nadjahi et al., 2019]"}],"fun_headline_variants":["Relation-aware slicing: cheaper, sharper GW alignment","Pairwise relations pick better slices for GW distance","Optimization-free relation-aware slicing for Gromov-Wasserstein","Data-guided slicing improves cross-domain alignment without tweaks","RASGW: slicing with pairwise hints beats uniform sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The claimed statistical rate rests on the assumption that the worst one-dimensional projection of the Gromov-Wasserstein distance is bounded by a fixed constant times the worst one-dimensional projection of the ordinary Wasserstein distance; if that inequality holds only under extra conditions, the $O(n^{-1/2})$ guarantee is not established.","fun_headline_variants_meta":{"raw":{"variants":["Relation-aware slicing: cheaper, sharper GW alignment","Pairwise relations pick better slices for GW distance","Optimization-free relation-aware slicing for Gromov-Wasserstein","Data-guided slicing improves cross-domain alignment without tweaks","RASGW: slicing with pairwise hints beats uniform sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000238,"raw_usage":{"total_tokens":1557,"prompt_tokens":1039,"completion_tokens":518,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":440}},"tokens_in":655,"tokens_out":518,"duration_ms":6417,"temperature":1.0,"reasoning_tokens":440,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:28:38.748751+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the ratio $\\mathrm{Max\\text{-}SGW}_2(\\mu_n,\\mu)/\\mathrm{Max\\text{-}SW}_2(\\mu_n,\\mu)$ on a compactly supported two-cluster Gaussian in $\\mathbb{R}^3$ as $n$ grows; Proposition 4.9 requires this ratio to stay bounded by a universal constant, so a ratio that grows with $n$ or with cluster separation would falsify the claimed statistical rate.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces sliced Gromov-Wasserstein via Radon projections and supplies the proof template for zero-distance implies isometric isomorphism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the inequality that bounds max-sliced Gromov-Wasserstein by a constant times max-sliced Wasserstein, underpinning the sample complexity proof of Proposition 4.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the power spherical distribution and its differentiable sampling, used for reparameterized gradient estimation."}],"review_version":1}