{ "schemaVersion": "1.6", "slug": "sinkhorn-metric-curvature-bias", "title": "Extrinsic curvature bias in the small-regularisation limit of Sinkhorn metrics", "shortTitle": "Sinkhorn metrics on curved data: the extrinsic curvature bias", "url": "https://evidencepress.org/releases/sinkhorn-metric-curvature-bias/", "oneLine": "On a curved surface sitting in space, with straight-line transport cost, the Sinkhorn metric converges to the Wasserstein metric as the regularisation ε shrinks, but at small ε it reads short by (ε/4) times a term measuring how the surface bends in space: on a circle of radius R every mode is multiplied by 1 − ε/(4R²), and a cheap positive correction removes the error.", "abstract": "Entropic (Sinkhorn) optimal transport defines a local Riemannian metric on probability distributions, and as the regularisation ε tends to 0 it should recover Wasserstein geometry. For a smooth positive density on a compact connected submanifold of Euclidean space without boundary, with the fixed ambient cost |x−y|², this candidate proves that the Sinkhorn metric operator converges in operator norm on centred L² to the inverse weighted Laplacian, at rate O(ε^{1/2}) in L² operator norm and O(ε) from H² to L². For a smooth tangent hμ with Aφ = h and A = −ρ⁻¹div(ρ∇), the metric satisfies g_ε(hμ,hμ) = ∫|∇φ|²dμ − (ε/4)∫Σ_α|S_α∇φ|²dμ + O(ε²), where S_α are the shape operators of the embedding. The first bias is therefore nonpositive and depends on the embedding, not only on the intrinsic geometry: the transport cost uses the chord, which is shorter than the arc. On a circle of radius R each eigenvalue is multiplied by 1 − ε/(4R²) to first order, and this order is sharp. A positive correction built from the entropic inverse removes the term; the expansions hold along smooth finite-dimensional density families; round spheres are computed exactly with an explicit ε² coefficient; a high-frequency family on the circle shows that no operator-norm rate better than O(ε) is possible; and isometric embeddings, such as a round circle and an ellipse of equal length, give different biases. The formal limit is due to Lavenant, Luckhardt, Mordant, Schmitzer and Tamanini; operator-level curvature expansions of Gaussian kernels are credited to Smolyanov, von Weizsäcker and Wittich, Liou and Lu, and Guo, whose full text could not be read. Population densities on closed manifolds only. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-04", "dateModified": "2026-11-04", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23940148", "doiUrl": "https://doi.org/10.5281/zenodo.23940148", "conceptDoi": "10.5281/zenodo.23940147", "pdfUrl": "https://github.com/ipitchford/sinkhorn-metric-curvature-bias/releases/download/v0.1.0-candidate/sinkhorn-metric-curvature-bias-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23940148", "repoUrl": "https://github.com/ipitchford/sinkhorn-metric-curvature-bias", "releaseUrl": "https://github.com/ipitchford/sinkhorn-metric-curvature-bias/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/sinkhorn-metric-curvature-bias/index.md", "bibtexUrl": "https://evidencepress.org/releases/sinkhorn-metric-curvature-bias/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/sinkhorn-metric-curvature-bias.mp3", "imageUrl": "https://evidencepress.org/assets/og/sinkhorn-metric-curvature-bias.png?v=f95d0187be", "coverArtUrl": "https://evidencepress.org/assets/art/sinkhorn-metric-curvature-bias.svg?v=f5bb7be87d", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/sinkhorn-metric-curvature-bias.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/sinkhorn-metric-curvature-bias.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate in optimal transport and geometric analysis with written proofs; numerical checks are floating-point checks only. Internal AI-assisted review in five roles, a separate-model adversarial review, a clean replay, one revision round with extensions, a re-review, a proof audit and a separate-model audit of the new proofs. 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/23940148", "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": "All seven scripts were rerun with Python -I -B, with and without -O, from fresh copies of the approved package; the symbolic, sphere/torus and torus-of-revolution outputs reproduced byte for byte and every other number agreed to roundoff (relative 1.4e-8 or better; residual-level fields near 1e-14 differ at roundoff). The frozen archive was bound to that replay, the symbolic and sphere checks were rerun from it under -O, and a corrupted circle coefficient was rejected; GitHub CI reruns all seven on Linux. These are floating-point checks, not proofs." }, { "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": "not-assessed", "note": "No independent reimplementation has been established; the package's independent circle and ellipse scripts are internal." }, { "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; the theorems rest on written proofs." }, { "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": "Scripts pin NumPy 2.3.5, SciPy 1.17.0, Matplotlib 3.10.8, mpmath 1.3.0 and SymPy 1.14.0 (requirements.txt); not a locked container; outputs reproduce to roundoff across Python and BLAS builds." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research, proof writing and numerical checks", "AI-assisted internal review in five roles (Claude Code subagents); adversarial review and audit of new proofs by a separate model (OpenAI Codex); clean replay; re-review and proof audit", "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 4 November 2026 with DOI insertion, 23 pages), its check scripts, and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 4 November 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Sinkhorn divergence", "entropic optimal transport", "Riemannian metric", "Wasserstein geometry", "shape operator", "extrinsic curvature", "embedded manifolds", "small-regularisation limit" ], "keyResults": [ "Limit and bias (Theorem 2.1): the Sinkhorn metric operator converges in operator norm to the inverse weighted Laplacian, and g_ε(hμ,hμ) = ∫|∇φ|²dμ − (ε/4)∫⟨Q∇φ,∇φ⟩dμ + O(ε²) with Q = Σ_α S_α², the squared shape operators: a nonpositive first bias that depends on the embedding.", "Rates (Theorem 2.4): ‖B_ε − A⁻¹‖ = O(ε^{1/2}) in L² operator norm and O(ε) from H² to L²; on the circle no L² operator-norm rate better than O(ε) is possible (Proposition 9.2); whether O(ε) holds in general is open.", "Correction and families (Corollary 2.2, Theorem 2.3): a positive corrected metric built from the entropic inverse is accurate to O(ε²) on fixed smooth directions, and all expansions hold along smooth finite-dimensional density families.", "Exact cases (Section 9.1, Proposition 9.1): on a circle of radius R each eigenvalue R²/k² becomes (R²/k²)(1 − ε/(4R²)) to first order, so on the unit circle 1/k² becomes (1/k²)(1 − ε/4), and this order is sharp; on round spheres S^m the metric is computed exactly with an explicit ε² coefficient.", "Embedding dependence (Proposition 10.1): isometric embeddings give different first-order biases; a round circle and an ellipse of the same length 2π give coefficients −0.125 and −0.043 for φ = cos s, and no closed embedded manifold has Q ≡ 0." ], "reviews": [], "evidencePackage": "Every theorem has a written proof: Theorems 2.1, 2.3 and 2.4, Corollary 2.2, Lemmas 3.1, 4.1, 5.1, 6.1, 7.1 and 8.1, and Propositions 5.2, 6.2, 9.1, 9.2 and 10.1. Seven scripts give checks: a symbolic check of the sphere coefficients, 60-digit sphere eigenvalues for S¹ to S⁴, a Funk–Hecke check, the flat torus in R⁴, a torus of revolution at nonconstant density, an equal-length ellipse against the circle, circle operator-norm evidence, a nonuniform ellipse at 768 and 1536 nodes, and independent circle and ellipse implementations. All seven were replayed with and without Python -O from fresh copies; outputs matched the shipped results to floating-point roundoff; a corrupted circle coefficient is rejected under -O; GitHub CI reruns all seven on Linux. The numerical checks are not certificates or proofs.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Lavenant, Luckhardt, Mordant, Schmitzer and Tamanini (2025). The Riemannian geometry of Sinkhorn divergences. Annales de l'Institut Henri Poincaré C, Analyse non linéaire. Introduces the Sinkhorn metric and its formal small-regularisation limit, made rigorous here for embedded manifolds.", "url": "https://doi.org/10.4171/aihpc/165", "doi": "10.4171/aihpc/165" }, { "citation": "Mordant and Xu (2026). Convergence of the Sinkhorn Riemannian metric for finitely supported measures. arXiv:2609.14651. The limit for finite supports; this paper treats continuous densities.", "url": "https://arxiv.org/abs/2609.14651", "doi": "10.48550/arXiv.2609.14651" }, { "citation": "Navas (2026). The information geometry of the Sinkhorn divergence on a Hilbert manifold. arXiv:2610.39850. Poses the continuous-density limit as an open direction.", "url": "https://arxiv.org/abs/2610.39850", "doi": "10.48550/arXiv.2610.39850" }, { "citation": "Lavenant, Luckhardt and Schmitzer (2026). Approximation and computation of the geodesic Sinkhorn distance. arXiv:2610.02007. Computation of Sinkhorn geodesics.", "url": "https://arxiv.org/abs/2610.02007", "doi": "10.48550/arXiv.2610.02007" }, { "citation": "Schubr (2026). Leading bias and positive correction of heat-kernel Sinkhorn metrics (Edition 2). Luminary Research Archive. The closest analogue, for an intrinsic heat-kernel cost.", "url": "https://research.luminaryoddities.com/leading-bias-positive-correction-heat-kernel-sinkhorn-metrics/v/2", "doi": null }, { "citation": "Schubr (2026). Sharp convergence of heat-kernel Sinkhorn metrics (Edition 2). Luminary Research Archive. An operator-norm rate for the heat-kernel cost.", "url": "https://research.luminaryoddities.com/sharp-convergence-of-heat-kernel-sinkhorn-metrics/v/2", "doi": null }, { "citation": "Guo (2026). Intrinsic and Extrinsic Bias of Diffusion Operators: Asymptotic Expansion and Kernel-Dependent Comparison. SSRN. Operator-level intrinsic and extrinsic bias expansions; only the abstract could be read.", "url": "https://doi.org/10.2139/ssrn.7411565", "doi": "10.2139/ssrn.7411565" }, { "citation": "Jia-Ming Liou and Chih-Chung Lu (2025). Asymptotic expansion of the Gaussian integral operators on Riemannian submanifolds of R^n. arXiv:2506.13238. General Gaussian expansions on submanifolds, credited.", "url": "https://arxiv.org/abs/2506.13238", "doi": "10.48550/arXiv.2506.13238" }, { "citation": "Smolyanov, von Weizsäcker and Wittich (2004). Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds. arXiv:math/0409155. Early second-fundamental-form terms in Gaussian kernel expansions, credited.", "url": "https://arxiv.org/abs/math/0409155", "doi": null }, { "citation": "Coifman and Lafon (2006). Diffusion maps. Applied and Computational Harmonic Analysis. 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Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Springer. Funk–Hecke formula used for the exact sphere computation.", "url": "https://doi.org/10.1007/978-3-642-25983-8", "doi": "10.1007/978-3-642-25983-8" }, { "citation": "Anonymous (2026). Curvature, relative entropy, and global Sinkhorn convergence. Evidence Press candidate (Zenodo). 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