{ "schemaVersion": "1.6", "slug": "gaussian-gromov-wasserstein-exactness", "title": "Affine rigidity and Gaussian exactness in squared-distance Gromov–Wasserstein transport", "shortTitle": "Twelve, or under 76/7: when the Gaussian shortcut for Gromov–Wasserstein is exact", "url": "https://evidencepress.org/releases/gaussian-gromov-wasserstein-exactness/", "oneLine": "Matching a standard Gaussian plane to a standard Gaussian line, the closed-form Gaussian Gromov–Wasserstein answer is 12, but a curved rearrangement that keeps the line exactly Gaussian provably scores below 76/7 ≈ 10.86; the paper shows the Gaussian shortcut is exact if and only if the two spectra are proportional, and is never off by more than a factor of two.", "abstract": "Gromov–Wasserstein (GW) transport compares two datasets or shapes that may live in different spaces by matching their internal pairwise distances. It is expensive to compute, so a closed-form answer for Gaussian data, found by Delon, Desolneux and Salmona (2022) after restricting attention to Gaussian couplings, is an attractive shortcut; those authors conjectured that the restriction loses nothing, and Peyré (October 2026) gave a first counterexample in sufficiently high dimension. For the standard squared-distance objective this candidate shows exactly when the shortcut is right: two Gaussians of positive rank admit an optimal coupling that is itself Gaussian if and only if their covariances have the same rank and proportional ordered eigenvalues. In every other case, including equal dimensions, an explicit nonlinear coupling that keeps both marginals exactly Gaussian does strictly better. The Gaussian excess is bracketed on both sides by quantities computable from the two spectra, and the Gaussian formula never overstates the optimum by more than a factor of two, a constant approached by nearly isotropic spectra in high dimension. For a standard Gaussian plane matched to a standard Gaussian line, the Gaussian answer is 12, an explicit coupling is certified below 76/7 ≈ 10.86 (about 10.47 numerically), and the true optimum lies between 8.68 and 10.85. Necessity comes from a general affine rigidity theorem: for a source whose support has interior points, an affine map is GW-optimal onto its image only if it is a similarity. The plug-in Gaussian formula on sample covariances is an inconsistent estimator for nonproportional spectra. The unrestricted optimum for nonproportional spectra is not computed. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-01", "dateModified": "2026-11-01", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23850758", "doiUrl": "https://doi.org/10.5281/zenodo.23850758", "conceptDoi": "10.5281/zenodo.23850757", "pdfUrl": "https://github.com/ipitchford/gaussian-gromov-wasserstein-exactness/releases/download/v0.1.0-candidate/gaussian-gromov-wasserstein-exactness-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23850758", "repoUrl": "https://github.com/ipitchford/gaussian-gromov-wasserstein-exactness", "releaseUrl": "https://github.com/ipitchford/gaussian-gromov-wasserstein-exactness/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/gaussian-gromov-wasserstein-exactness/index.md", "bibtexUrl": "https://evidencepress.org/releases/gaussian-gromov-wasserstein-exactness/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/gaussian-gromov-wasserstein-exactness.mp3", "imageUrl": "https://evidencepress.org/assets/og/gaussian-gromov-wasserstein-exactness.png?v=b1f3d93caf", "coverArtUrl": "https://evidencepress.org/assets/art/gaussian-gromov-wasserstein-exactness.svg?v=270564779e", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/gaussian-gromov-wasserstein-exactness.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/gaussian-gromov-wasserstein-exactness.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted optimal-transport candidate. The theorems have written proofs; identities and finite cases are checked exactly and examples numerically; values marked numerical are not interval enclosures. Internal AI-assisted review in five roles, a separate-model adversarial review and audit of new proofs, a separate proof audit, a clean replay of the handoff bundle, one revision round and a re-review. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23850758", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "From a fresh extraction of the frozen archive, all seven checking scripts ran with Python -I -O -B in a temporary copy and every regenerated output matched the retained one within a relative tolerance of 1e-7, apart from run-time and interpreter fields; negative controls detected 12 of 12 planted errors. GitHub CI repeats the quick checks on Linux as a blocking job (passing) and the full suite as a non-blocking job: on Linux every check in the full suite passes, but seven numerical-error diagnostic fields (finite-difference residuals and quadrature-error estimates) differ from the macOS values by more than the comparer's 1e-7 relative tolerance, so that job reports a difference." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "A separately written rational-polynomial audit re-derives 28 identities, and independent high-precision methods agree on the comonotone integral. This is producer-side diversity, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Python 3.12 with NumPy, SciPy, Matplotlib and mpmath versions pinned in requirements.txt; not a locked container. The exact checkers need only the standard library." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted optimal-transport mathematics", "AI-assisted internal review, separate-model adversarial review and audit, proof audit, re-review", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript (revision of 1 November 2026 with DOI insertion, 16 pages), its checking scripts and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 1 November 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Gromov–Wasserstein", "optimal transport", "Gaussian measures", "Gaussian coupling", "affine rigidity", "linear Gromov–Monge", "measure-preserving flow", "approximation error" ], "keyResults": [ "Exactness (Theorem 5, proof): two positive-rank Gaussians admit a jointly Gaussian unrestricted optimiser of squared-distance GW if and only if their positive ranks agree and their ordered positive covariance eigenvalues are proportional; the conjecture of Delon, Desolneux and Salmona fails on every other pair, including equal dimensions. Sufficiency is due to Delon et al.", "Affine rigidity (Theorem 3, proof): for a source with finite fourth moment and support with interior points in its affine hull, an affine map is GW-optimal onto its image if and only if it is a similarity or a constant.", "Size of the error (Theorem 8, Corollary 9, proof): with S = ‖a‖²‖b‖² − (a·b)², S/(‖a‖‖b‖) ≤ Q_G − Q_* ≤ 8(‖a‖‖b‖ − a·b), so Q_G/2 ≤ Q_* ≤ Q_G, with the factor two approached by nearly isotropic spectra in high dimension. The upper bound is due to Delon et al.; the lower bound is new.", "Plane to line (Section 7, proof with numerical refinements): the Gaussian value is 12; an explicit Gaussian-preserving coupling scores below 76/7 ≈ 10.86 (about 10.47 numerically); 8.68 ≤ Q_* ≤ 10.85, refined numerically to about 8.82 ≤ Q_* ≤ 10.03.", "Plug-in inconsistency (Lemma 11, Proposition 12, proof): for nonproportional spectra the empirical GW value converges to Q_* while the plug-in Gaussian formula on sample covariances converges to Q_G ≠ Q_*." ], "reviews": [], "evidencePackage": "Written proofs of every theorem, proposition, lemma and corollary; seven checking scripts with retained outputs: 24 exact rational identities including the certificate Q < 76/7 and all 441 sorted two-coordinate integer spectrum pairs with entries 0 to 5, 28 identities in a separately written rational-polynomial implementation, simultaneous-field identities on 120 integer spectra, the four-point example and flow reductions, the factor-two corollary on 1,156 integer and 20,000 random pairs, whole-path curvature checks, Tables 1 and 2 and quadrature of the plane-to-line flow; a negative-control script detecting 12 of 12 planted errors with and without Python -O; replay with Python -O from a fresh extraction of the frozen archive.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Delon, Desolneux and Salmona (2022). Gromov–Wasserstein distances between Gaussian distributions. Journal of Applied Probability. The Gaussian-restricted value, sufficiency of proportional spectra, the upper bound on the excess and the conjecture resolved here.", "url": "https://doi.org/10.1017/jpr.2022.16", "doi": "10.1017/jpr.2022.16" }, { "citation": "Peyré (2026). Gaussians are not closed for Gromov–Wasserstein. Blog post, 23 October 2026. The first counterexample, for a line against a sufficiently high-dimensional space.", "url": "https://www.gpeyre.com/blog/2026/10/23/gaussian-gromov-wasserstein-counterexample/", "doi": null }, { "citation": "Peyré (2026). Optimal Transport for Machine Learners. Online book, Propositions 13.33–13.34.", "url": "https://www.gpeyre.com/ot4ml/OT4ML/OT4ML.pdf", "doi": null }, { "citation": "Salmona (2023; Crossref DOI record created 2026). Transport of probability distributions across different Euclidean spaces. Doctoral thesis, Université Paris-Saclay. Describes the construction of counterexamples as open.", "url": "https://doi.org/10.70675/a8e13508z4193z4651za1f4z4f4199ba110c", "doi": "10.70675/a8e13508z4193z4651za1f4z4f4199ba110c" }, { "citation": "Mémoli (2011). Gromov–Wasserstein Distances and the Metric Approach to Object Matching. Foundations of Computational Mathematics. GW distances for metric-measure spaces.", "url": "https://doi.org/10.1007/s10208-011-9093-5", "doi": "10.1007/s10208-011-9093-5" }, { "citation": "Dumont, Lacombe and Vialard (2025; online 2024). On the Existence of Monge Maps for the Gromov–Wasserstein Problem. Foundations of Computational Mathematics. The linearisation used in the affine rigidity theorem.", "url": "https://doi.org/10.1007/s10208-024-09643-0", "doi": "10.1007/s10208-024-09643-0" }, { "citation": "Zhang, Goldfeld, Mroueh and Sriperumbudur (2024). Gromov–Wasserstein distances: Entropic regularization, duality and sample complexity. The Annals of Statistics. Variational form of squared-distance GW.", "url": "https://doi.org/10.1214/24-AOS2406", "doi": "10.1214/24-AOS2406" }, { "citation": "Vayer (2020). A contribution to Optimal Transport on incomparable spaces. Doctoral thesis, arXiv:2011.04447. A sufficient radial condition for optimal maps.", "url": "https://arxiv.org/abs/2011.04447v1", "doi": "10.48550/arXiv.2011.04447" }, { "citation": "Morel, Drumetz, Benaïchouche, Courty and Rousseau (2023). Turning Normalizing Flows into Monge Maps with Geodesic Gaussian Preserving Flows. Transactions on Machine Learning Research; arXiv:2209.10873. Gaussian-preserving rearrangements in ordinary optimal transport.", "url": "https://arxiv.org/abs/2209.10873v4", "doi": "10.48550/arXiv.2209.10873" }, { "citation": "Arya, Auddy, Clark, Lim, Mémoli and Packer (2026; online 2024). The Gromov–Wasserstein Distance Between Spheres. Foundations of Computational Mathematics. A sphere result with a Gaussian subproblem.", "url": "https://doi.org/10.1007/s10208-024-09678-3", "doi": "10.1007/s10208-024-09678-3" }, { "citation": "Le, Le, Nguyen, Do, Pham and Ho (2022). Entropic Gromov-Wasserstein between Gaussian Distributions. Proceedings of the 39th International Conference on Machine Learning (PMLR 162). Entropic and inner-product variants, a different objective.", "url": "https://proceedings.mlr.press/v162/le22a.html", "doi": null }, { "citation": "Dandapanthula, Podkopaev, Kasiviswanathan, Ramdas and Goldfeld (2026). Optimal Transportation and Alignment Between Gaussian Measures. arXiv:2512.03579, version 2.", "url": "https://arxiv.org/abs/2512.03579v2", "doi": "10.48550/arXiv.2512.03579" }, { "citation": "Tran and Soh (2026). Quadratic Optimization over Probability Measures with Coupling Constraints. arXiv:2608.24184. Moment–sum-of-squares relaxations; states that the general Gaussian value is not known.", "url": "https://arxiv.org/abs/2608.24184v1", "doi": "10.48550/arXiv.2608.24184" }, { "citation": "Van Assel, Vincent-Cuaz, Courty, Flamary, Frossard and Vayer (2025). Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein. arXiv:2402.02239, version 3.", "url": "https://arxiv.org/abs/2402.02239v3", "doi": "10.48550/arXiv.2402.02239" }, { "citation": "Flamary, Courty, Gramfort et al. (2021). POT: Python Optimal Transport. Journal of Machine Learning Research 22(78). Its default square-loss GW objective is the one studied here.", "url": "https://jmlr.org/papers/v22/20-451.html", "doi": null }, { "citation": "Villani (2009). Optimal Transport: Old and New. Springer, Grundlehren der mathematischen Wissenschaften 338.", "url": "https://doi.org/10.1007/978-3-540-71050-9", "doi": "10.1007/978-3-540-71050-9" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:gaussian-gromov-wasserstein-exactness", "attemptIds": [ "ep-attempt:gaussian-gromov-wasserstein-exactness-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "decision-rule", "description": "Use the closed-form Gaussian GW value as the true squared-distance GW value only when the two covariance spectra have equal positive rank and are proportional; otherwise report the certified bracket Q_G − 8(‖a‖‖b‖ − a·b) ≤ Q_* ≤ Q_G − S/(‖a‖‖b‖), which always lies within a factor of two.", "scope": "Squared-distance GW with squared loss between Gaussian marginals. Non-Gaussian data, entropic, inner-product and ordinary-distance GW are not covered." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Section 2 restates the moment reduction and Gaussian-restricted value of Delon et al. in the paper's normalisation, and Section 1.3 separates known from new results, using the numbering of arXiv:2104.07970v1.", "remainingRisks": [ "The bounded literature check is not exhaustive priority certification.", "Numerical values are not interval enclosures." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Have a specialist in optimal transport check the cyclical-monotonicity step of Theorem 3 and the whole-path curvature bound behind Theorem 8, and ask the authors of the Gaussian GW paper whether Theorem 5 matches the intended scope of their conjecture.", "claimCeiling": "Proved classification of Gaussian exactness for squared-distance GW, an affine rigidity theorem and two-sided error bounds. Internal verification is not external assurance." }, "impactClaims": [ { "id": "gaussian-gromov-wasserstein-exactness-no-impact", "aim": "science", "outcome": "An openly inspectable classification of when the closed-form Gaussian GW value is exact, with certified error brackets.", "setting": "AI-assisted mathematical research", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "None registered.", "estimand": "No research-speed or scientific-impact effect is estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ] }, "researchMetrics": { "schemaVersion": "1.0", "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json", "outcomePolicyUrl": "https://evidencepress.org/api/v2/research-metrics-policy.json", "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json", "claimCeiling": "These records describe scoped workflow inputs and outcomes. They support calibration and reference-class construction, but do not by themselves establish research acceleration, theorem correctness, novelty, independent assurance or impact.", "attempts": [ { "attemptId": "ep-attempt:gaussian-gromov-wasserstein-exactness-publication", "status": "active", "resultClass": "pending", "metrics": { "schemaVersion": "1.0", "measurementScope": "publication-only", "scopeBoundary": "Only archive packaging, site integration, guarded delivery and canonical readback from registration are prospective. Earlier research, review, revision and extension work are left-censored.", "forecast": { "frozenAt": "2026-11-01T01:29:00Z", "procedureClass": "full-candidate-publication", "targetOutcome": "Accurately scoped GitHub, Zenodo and canonical Evidence Press release with explanatory media and machine readback, if all publication gates pass.", "expectedActiveMinutes": 90, "plausibleLowMinutes": 60, "plausibleHighMinutes": 150, "expectedUnattendedWaitMinutes": 30, "referenceClass": { "label": "Full mathematical candidate publication", "sampleSize": 0, "basis": "Route benchmark planning prior, not empirical calibration." }, "fermiComponents": [ { "component": "Research archive packaging, replay and identities", "count": 1, "lowMinutesPerUnit": 25, "centralMinutesPerUnit": 35, "highMinutesPerUnit": 60, "basis": "One frozen package with a rating-free public export, clean replay and DOI insertion." }, { "component": "Page, metadata and media integration", "count": 1, "lowMinutesPerUnit": 20, "centralMinutesPerUnit": 30, "highMinutesPerUnit": 45, "basis": "One reader-first release, drafted in parallel with verification." }, { "component": "Guarded deployment and canonical readback", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 45, "basis": "Mandatory site tests, one merge cycle with ledger carry-forward, zero-cost deployment and public checks." } ], "probabilityPositiveSignal": 0.9, "probabilityTargetClosure": 0.85, "probabilityHorizonMinutes": 300, "assumptions": [ "The approved scientific package remains valid and configured providers remain available." ], "reforecastTriggers": [ "A publication gate reveals a scientific or integrity defect, or a provider outage blocks delivery." ], "stopRule": "Do not weaken assurance, integrity or cost gates to meet the forecast." }, "outcome": null }, "missingnessReason": null } ] } }