pith. sign in

arxiv: 2606.23037 · v1 · pith:D5OZTV2Inew · submitted 2026-06-22 · 🧮 math.MG · math.FA

Two-mode stability for multi-marginal optimal transport maps

Bang-Xian Han , Zhuo-Nan Zhu This is my paper

Pith reviewed 2026-06-26 06:19 UTC · model grok-4.3

classification 🧮 math.MG math.FA
keywords multi-marginal optimal transportMonge solutionsstability estimatesWasserstein barycenterKantorovich defectHolder continuitybarycentric quadratic cost
0
0 comments X

The pith

Multi-marginal optimal transport maps with barycentric quadratic cost split into external barycentric and internal relative modes that admit separate stability estimates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a two-mode stability theory for Monge solutions of the multi-marginal optimal transport problem with barycentric quadratic cost. The tuple of maps decomposes into an external mode given by the transport map to the Wasserstein barycenter and internal relative modes. A quadratic lower bound on the Kantorovich defect controls the internal modes to produce a square-root stability estimate without using two-marginal map stability results. This decomposition yields a 1/4-Holder estimate under general perturbations and a 1/2-Holder estimate under barycenter-preserving perturbations, with both exponents shown to be optimal along with the dependence on weights. The paper also determines the range of costs beyond the barycentric case for which the two-mode estimate continues to hold.

Core claim

For Monge solutions of the multi-marginal optimal transport problem with barycentric quadratic cost, the associated tuple of maps splits into an external barycentric mode and internal relative modes. The external mode coincides with the optimal transport map from the fixed source to the Wasserstein barycenter. A quadratic lower bound for the Kantorovich defect controls the internal modes and yields a square-root estimate without invoking any two-marginal map-stability theorem. Combining the two-mode estimate with a sharp two-marginal stability result produces a 1/4-Holder estimate for general perturbations and a 1/2-Holder estimate for barycenter-preserving perturbations, and both exponents

What carries the argument

Two-mode decomposition of the tuple of maps into an external barycentric mode and internal relative modes, controlled by a quadratic lower bound on the Kantorovich defect for the internal modes.

If this is right

  • General perturbations produce a 1/4-Holder estimate on the maps.
  • Barycenter-preserving perturbations produce a 1/2-Holder estimate on the maps.
  • Both exponents and the dependence on the weights are optimal.
  • The two-mode estimate extends to collective-coordinate perturbations and to uniformly concave costs of the sum.
  • For graph interactions, hedonic costs, and translation-invariant costs the decomposition fails because of loss of relative coercivity or lack of stability for the external modes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mode separation could reduce the computational cost of solving multi-marginal problems by treating the barycenter transport independently from relative adjustments.
  • Numerical checks on low-dimensional examples with barycentric quadratic cost could directly verify the square-root scaling of internal-mode errors against the Kantorovich defect.
  • The same splitting idea may be tested on costs that preserve a collective coordinate but are not strictly quadratic.

Load-bearing premise

The tuple of maps decomposes into an external barycentric mode and internal relative modes for the barycentric quadratic cost.

What would settle it

A concrete multi-marginal instance with barycentric quadratic cost in which the Kantorovich defect fails to quadratically bound the deviation of the internal relative modes.

read the original abstract

We establish a two-mode stability theory for Monge solutions of the multi-marginal optimal transport problem with barycentric quadratic cost. The associated tuple of maps splits into an external barycentric mode and internal relative modes. A quadratic lower bound for the Kantorovich defect controls the internal modes and yields a square-root estimate without invoking any two-marginal map-stability theorem. The external mode is the optimal transport map from the fixed source to the Wasserstein barycenter. Combining the resulting two-mode estimate with M\'erigot's sharp theorem gives a $\frac{1}{4}$-H\"older estimate for general perturbations, while barycenter-preserving perturbations satisfy a $\frac{1}{2}$-H\"older estimate. We prove that both exponents and the dependence on the weights are optimal. We also examine the scope of such decomposition beyond the barycentric cost: collective-coordinate perturbations and uniformly concave costs of the sum retain the two-mode estimate, whereas the analyses of graph interactions, hedonic costs, and translation-invariant costs identify the two possible obstructions -- loss of relative coercivity and lack of stability for the remaining external modes.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes a two-mode stability theory for Monge solutions of the multi-marginal optimal transport problem with barycentric quadratic cost. The tuple of maps decomposes into an external barycentric mode (the OT map to the Wasserstein barycenter) and internal relative modes. A quadratic lower bound on the Kantorovich defect controls the internal modes to produce a square-root stability estimate without invoking two-marginal map-stability theorems. Combining this with Mérigot's theorem yields a 1/4-Hölder estimate for general perturbations and a 1/2-Hölder estimate for barycenter-preserving perturbations; both exponents and the weight dependence are shown to be optimal. The scope of the decomposition is examined for other costs, identifying obstructions such as loss of relative coercivity and lack of external-mode stability.

Significance. If the central two-mode decomposition and quadratic defect bound hold, the work supplies a new structural approach to multi-marginal stability that bypasses two-marginal results and delivers sharp Hölder exponents with optimality proofs. The analysis of which costs retain the decomposition (collective-coordinate perturbations, uniformly concave sum costs) versus those that do not (graph interactions, hedonic costs, translation-invariant costs) clarifies the boundary of the technique.

minor comments (3)
  1. The abstract states that the decomposition 'splits into an external barycentric mode and internal relative modes' and that the quadratic lower bound 'controls the internal modes'; the corresponding section should explicitly state the precise functional setting (e.g., the measures, the cost function, and the precise definition of the Kantorovich defect) before the decomposition is invoked.
  2. When the two-mode estimate is combined with Mérigot's theorem, the dependence on the weights should be tracked explicitly through the constants; the optimality claim for the weight dependence would be strengthened by a short remark indicating where this dependence appears in the final Hölder constants.
  3. The discussion of obstructions for non-barycentric costs (loss of relative coercivity, lack of stability for external modes) would benefit from a short table or bullet list summarizing which costs fall into each category.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a two-mode decomposition of Monge maps into external barycentric and internal relative modes specifically for the barycentric quadratic cost, derives a quadratic lower bound on the Kantorovich defect controlling the internal modes, and obtains square-root estimates without invoking two-marginal stability results. This structure is then combined with the external Mérigot theorem to obtain Hölder exponents, with optimality claims for the exponents and weights. No equations, fitted parameters, self-citations, or ansatzes are exhibited that reduce any claimed prediction or estimate to an input by construction; the central derivation is self-contained against the stated assumptions and external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or verified from the provided text.

pith-pipeline@v0.9.1-grok · 5723 in / 1169 out tokens · 29279 ms · 2026-06-26T06:19:00.905823+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 4 canonical work pages

  1. [1]

    Martial Agueh and Guillaume Carlier,Barycenters in the Wasserstein space, SIAM J. Math. Anal.43(2011), no. 2, 904–924. 3, 14, 16, 21, 27

    2011

  2. [2]

    2062, Springer, Heidelberg, 2013, pp

    Luigi Ambrosio and Nicola Gigli,A user’s guide to optimal transport, in: Modelling and optimisation of flows on networks, Lecture Notes in Math., vol. 2062, Springer, Heidelberg, 2013, pp. 1–155. 10 32

    2062

  3. [3]

    Berman,Convergence rates for discretized Monge–Amp` ere equations and quantitative stability of optimal transport, Found

    Robert J. Berman,Convergence rates for discretized Monge–Amp` ere equations and quantitative stability of optimal transport, Found. Comput. Math.21(2021), no. 4, 1099–1140. 2

    2021

  4. [4]

    J´ er´ emie Bigot, Elsa Cazelles and Nicolas Papadakis,Penalization of barycenters in the Wasserstein space, SIAM J. Math. Anal.51(2019), no. 3, 2261–2285, doi:10.1137/18M1185065. 27

    work page doi:10.1137/18m1185065 2019

  5. [5]

    Stat.22(2018), 35–57

    J´ er´ emie Bigot and Thierry Klein,Characterization of barycenters in the Wasserstein space by averaging optimal transport maps, ESAIM Probab. Stat.22(2018), 35–57. 21

    2018

  6. [6]

    Pure Appl

    Yann Brenier,Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math.44(1991), no. 4, 375–417. 2

    1991

  7. [7]

    Theory Related Fields188(2024), no

    Guillaume Carlier, Alex Delalande, and Quentin M´ erigot,Quantitative stability of barycenters in the Wasserstein space, Probab. Theory Related Fields188(2024), no. 3–4, 1257–1286. 14

    2024

  8. [8]

    Theory42(2010), no

    Guillaume Carlier and Ivar Ekeland,Matching for teams, Econom. Theory42(2010), no. 2, 397–418. 31

    2010

  9. [9]

    Guillaume Carlier and Bruno Nazaret,Optimal transportation for the determinant, ESAIM Control Optim. Calc. Var.14(2008), no. 4, 678–698. 32

    2008

  10. [10]

    Elsa Cazelles, ´Edouard Pauwels, and L´ eo Portales,Statistical estimation of Monge transport maps via Brenier potentials, arXiv:2604.22366 [math.OC], 2026. 2

    Pith/arXiv arXiv 2026

  11. [11]

    McCann, and Lars P

    Pierre-Andr´ e Chiappori, Robert J. McCann, and Lars P. Nesheim,Hedonic price equi- libria, stable matching, and optimal transport: equivalence, topology, and uniqueness, Econom. Theory42(2010), no. 2, 317–354. 3, 31

    2010

  12. [12]

    Maria Colombo, Simone Di Marino, and Federico Stra,Continuity of multimarginal optimal transport with repulsive cost, SIAM J. Math. Anal.51(2019), no. 4, 2903–2926. 3, 32

    2019

  13. [13]

    Pure Appl

    Codina Cotar, Gero Friesecke, and Claudia Kl¨ uppelberg,Density functional theory and optimal transportation with Coulomb cost, Comm. Pure Appl. Math.66(2013), no. 4, 548–599. 32

    2013

  14. [14]

    J.172(2023), no

    Alex Delalande and Quentin M´ erigot,Quantitative stability of optimal transport maps under variations of the target measure, Duke Math. J.172(2023), no. 17, 3321–3357. 2

    2023

  15. [15]

    Simone Di Marino, Augusto Gerolin, and Luca Nenna,Optimal transportation theory with repulsive costs, in: Topological optimization and optimal transport, Radon Ser. Comput. Appl. Math., vol. 17, De Gruyter, Berlin, 2017, pp. 204–256, doi:10.1515/9783110430417-010. 31, 32

    work page doi:10.1515/9783110430417-010 2017

  16. [16]

    Jin Feng,On a Hamilton–Jacobi PDE theory for hydrodynamic limit of action mini- mizing collective dynamics, arXiv:2512.20809v1 [math.AP], 2025. 29

    arXiv 2025

  17. [17]

    Pure Appl

    Wilfrid Gangbo and Andrzej ´Swi¸ ech,Optimal maps for the multidimensional Monge– Kantorovich problem, Comm. Pure Appl. Math.51(1998), no. 1, 23–45. 2, 3, 9, 15, 17, 18, 28 33

    1998

  18. [18]

    Augusto Gerolin, Anna Kausamo, and Tapio Rajala,Non-existence of optimal transport maps for the multi-marginal repulsive harmonic cost, SIAM J. Math. Anal.51(2019), no. 3, 2359–2371. 31

    2019

  19. [19]

    Nicola Gigli,On H¨ older continuity-in-time of the optimal transport map towards measures along a curve, Proc. Edinb. Math. Soc. (2)54(2011), no. 2, 401–409. 5

    2011

  20. [20]

    Henri Heinich,Probl` eme de Monge pournprobabilit´ es, C. R. Math. Acad. Sci. Paris 334(2002), no. 9, 793–795. 28

    2002

  21. [21]

    Kantorovitch,On the translocation of masses, C

    L. Kantorovitch,On the translocation of masses, C. R. (Dokl.) Acad. Sci. URSS37 (1942), 199–201. 2

    1942

  22. [22]

    Kellerer,Duality theorems for marginal problems, Z

    Hans G. Kellerer,Duality theorems for marginal problems, Z. Wahrscheinlichkeitstheo- rie verw. Gebiete67(1984), no. 4, 399–432. 9

    1984

  23. [23]

    Young-Heon Kim and Brendan Pass,A general condition for Monge solutions in the multi-marginal optimal transport problem, SIAM J. Math. Anal.46(2014), no. 2, 1538–1550. 24, 28

    2014

  24. [24]

    Young-Heon Kim and Brendan Pass,Multi-marginal optimal transport on Riemannian manifolds, Amer. J. Math.137(2015), no. 4, 1045–1060. 3, 9, 15, 16, 17, 18

    2015

  25. [25]

    Math.307(2017), 640–683

    Young-Heon Kim and Brendan Pass,Wasserstein barycenters over Riemannian mani- folds, Adv. Math.307(2017), 640–683. 16

    2017

  26. [26]

    G2, 333–344, doi:10.5802/crmath.834

    Cyril Letrouit,Unstable optimal transport maps, Comptes Rendus Math.364(2026), no. G2, 333–344, doi:10.5802/crmath.834. 2, 15, 17

    work page doi:10.5802/crmath.834 2026

  27. [27]

    Cyril Letrouit and Quentin M´ erigot,Gluing methods for quantitative stability of optimal transport maps, arXiv:2411.04908v2 [math.AP], 2025. 2

    arXiv 2025

  28. [28]

    2, 5, 13, 27

    Quentin M´ erigot,Sharp stability of Brenier maps via quantitative regularity of poten- tials, HAL preprint hal-05616391, 2026. 2, 5, 13, 27

    2026

  29. [29]

    Gaspard Monge,M´ emoire sur la th´ eorie des d´ eblais et des remblais, M´ em. Math. Phys. Acad. Royale Sci. (1781), 666–704. 2

  30. [30]

    Luca Nenna and Brendan Pass,An ODE characterization of multi-marginal optimal transport with pairwise cost functions, IMA J. Numer. Anal. (2025), doi:10.1093/imanum/draf067. 30

    work page doi:10.1093/imanum/draf067 2025

  31. [31]

    Brendan Pass,On the local structure of optimal measures in the multi-marginal optimal transportation problem, Calc. Var. Partial Differential Equations43(2012), no. 3–4, 529–536. 32

    2012

  32. [32]

    Brendan Pass,Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions, Discrete Contin. Dyn. Syst.34(2014), no. 4, 1623–1639. 3, 31

    2014

  33. [33]

    Brendan Pass,Multi-marginal optimal transport: theory and applications, ESAIM Math. Model. Numer. Anal.49(2015), no. 6, 1771–1790. 3, 30

    2015

  34. [34]

    3, 32 34

    Brendan Pass and Yair Shenfeld,A dynamical formulation of multi-marginal optimal transport, arXiv:2509.22494v2 [math.OC], 2025. 3, 32 34

    arXiv 2025

  35. [35]

    Old and new, Grundlehren der mathematischen Wissenschaften, vol

    C´ edric Villani,Optimal transport. Old and new, Grundlehren der mathematischen Wissenschaften, vol. 338, Springer, Berlin, 2009. 5 Declaration.The authors declare that they have no conflict of interest and that the manuscript has no associated data. F unding: This work was supported in part by the National Key R&D Program of China (2021YFA1000900, 2021YF...

    2009