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arxiv: 2604.15474 · v1 · submitted 2026-04-16 · 🧮 math.AP · math.OC

Instability of the ray-monotone selector for W₁-optimal transport

Maja Gwozdz This is my paper

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classification 🧮 math.AP math.OC
keywords Wasserstein distanceoptimal transportray-monotone plansstabilitycounterexampleweak convergenceGamma-convergencesecondary selection
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The pith

The ray-monotone selector for W1-optimal transport plans fails to be stable under weak convergence of targets.

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

The paper constructs a counterexample showing that the ray-monotone W1-optimal plan, obtained by secondary minimization of quadratic energy, does not remain stable when the target measure is perturbed by a weakly convergent sequence of absolutely continuous measures. For a fixed source measure mu that is absolutely continuous, targets nu_n converge weakly to nu, yet the selected plans for the sequence converge weakly to a different W1-optimal plan than the one selected directly for the limit pair. This negative result addresses an open question on continuity of the selector. The work also identifies the narrow Kuratowski limit of the sets of optimal plans and derives constrained Gamma-limits for families of secondary energies of the form integral Phi(|x-y|) d gamma.

Core claim

We construct a fixed absolutely continuous source mu and absolutely continuous targets nu_n that converge weakly to nu such that the selected plans gamma^sel(mu, nu_n) converge weakly to a plan gamma^hom that belongs to the set of W1-optimal plans O(mu, nu) but differs from the ray-monotone plan gamma^sel(mu, nu) itself.

What carries the argument

The ray-monotone W1-optimal plan obtained via secondary variational selection by the quadratic energy C2 among all plans optimal for the distance cost |x-y|.

If this is right

  • The narrow Kuratowski limit of the sets O(mu, nu_n) contains both the ray-monotone plan and the homogeneous plan.
  • Constrained Gamma-limits exist for secondary energies integral Phi(|x-y|) d gamma when Phi is any continuous function on [0,2].
  • The additive perturbation c_epsilon = |x-y| + epsilon |x-y|^2 yields a non-commutation result as epsilon tends to zero after the limit in n.

Where Pith is reading between the lines

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

  • Selection procedures in optimal transport that rely on quadratic secondary minimization can produce limits inconsistent with direct application to the limiting problem.
  • Numerical schemes approximating W1 plans via the ray-monotone selector may require additional regularization to ensure convergence under refinement of marginals.
  • The identified Kuratowski and Gamma limits suggest that the full set of optimal plans must be tracked rather than a single selected representative when passing to limits.

Load-bearing premise

Specific absolutely continuous measures exist in Euclidean space whose supports and densities make the quadratic secondary selection produce the ray-monotone plan and allow the stated weak and narrow convergences to hold.

What would settle it

Explicit construction and verification of the measures mu and nu_n such that either the limit of the selected plans equals the selected plan of the limit or the claimed weak convergence of selected plans fails to hold.

read the original abstract

For the distance cost $c(x,y)=|x-y|$, the set $O(\mu,\nu)$ of $W_1$-optimal plans is generally not a singleton. Under the classical absolute-continuity hypotheses in the Euclidean case, secondary variational selection by the quadratic energy $C_2$ yields the ray-monotone $W_1$-optimal plan. We provide a counterexample to an open problem posed by Santambrogio that concerns the stability of this selector under weak convergence of the marginals. More precisely, we construct a fixed absolutely continuous source $\mu$ and absolutely continuous targets $\nu_n\rightharpoonup\nu$ such that $\gamma^{\mathrm{sel}}(\mu,\nu_n)\rightharpoonup\gamma^{\mathrm{hom}}$, where $\gamma^{\mathrm{hom}}\in O(\mu,\nu)$ but $\gamma^{\mathrm{hom}}\neq\gamma^{\mathrm{sel}}(\mu,\nu)$. We also identify the narrow Kuratowski limit of the optimal-plan sets $O(\mu,\nu_n)$, derive the constrained $\Gamma$-limit for secondary energies of the form $\int \Phi(|x-y|)\,d\gamma$ with $\Phi\in C([0,2])$, and deduce a non-commutation result for the additive perturbation $c_\varepsilon(x,y)=|x-y|+\varepsilon|x-y|^2$.

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

2 major / 2 minor

Summary. The manuscript constructs a counterexample to the stability of the ray-monotone selector (equivalently, the C₂-secondary selector) for W₁-optimal transport plans. For a fixed absolutely continuous source μ and a sequence of absolutely continuous targets ν_n ⇀ ν, the selected plans γ^sel(μ, ν_n) converge weakly to a W₁-optimal plan γ^hom for the limit pair (μ, ν) that is distinct from γ^sel(μ, ν). The paper also identifies the narrow Kuratowski limit of the sets O(μ, ν_n), derives the constrained Γ-limit of secondary energies of the form ∫ Φ(|x-y|) dγ for Φ ∈ C([0,2]), and establishes non-commutation of the selector with the additive perturbation c_ε(x,y) = |x-y| + ε|x-y|².

Significance. The result directly resolves an open question of Santambrogio on stability of the ray-monotone selector under weak convergence of marginals, under the classical absolute-continuity hypotheses that guarantee uniqueness of the secondary selector. The explicit construction, together with the Kuratowski and Γ-limit statements, supplies concrete tools for analyzing non-uniqueness and selection in W₁ transport; the non-commutation corollary for perturbed costs is a natural byproduct. These contributions are of clear interest to the optimal-transport community.

major comments (2)
  1. [§3] §3 (construction of μ and ν_n): the explicit densities and supports must be verified to ensure that each ν_n is absolutely continuous, that the ray-monotone plan is indeed the C₂-selector for each (μ, ν_n), and that the stated weak convergence γ^sel(μ, ν_n) ⇀ γ^hom holds with γ^hom ≠ γ^sel(μ, ν). The abstract asserts these properties, but the load-bearing step is the verification that the limit plan is not ray-monotone for the limit pair; this requires checking the geometric conditions on the supports and the quadratic-energy comparison.
  2. [§4] §4 (narrow Kuratowski limit of O(μ, ν_n)): the identification of the limit set must be shown to be compatible with the weak convergence of the selected plans; if the Kuratowski limit contains plans other than γ^hom, the argument that γ^hom is the only possible weak limit point of the sequence γ^sel(μ, ν_n) needs an additional tightness or uniqueness argument.
minor comments (2)
  1. Notation: the symbol γ^hom is introduced without an explicit definition in the abstract; a sentence clarifying that it denotes a specific W₁-optimal plan (e.g., the one supported on a particular set of rays) would help readers.
  2. The statement of the Γ-limit result should specify the topology on the space of plans (narrow or weak) and the precise constraint set on which the Γ-convergence holds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our work, which resolves an open question of Santambrogio. We address each major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: §3 (construction of μ and ν_n): the explicit densities and supports must be verified to ensure that each ν_n is absolutely continuous, that the ray-monotone plan is indeed the C₂-selector for each (μ, ν_n), and that the stated weak convergence γ^sel(μ, ν_n) ⇀ γ^hom holds with γ^hom ≠ γ^sel(μ, ν). The abstract asserts these properties, but the load-bearing step is the verification that the limit plan is not ray-monotone for the limit pair; this requires checking the geometric conditions on the supports and the quadratic-energy comparison.

    Authors: We agree that explicit verification is necessary for the counterexample. The manuscript already supplies the explicit densities, supports, and main arguments in §3. In the revision we will add a dedicated verification subsection containing: (i) direct computation of the densities of each ν_n to confirm absolute continuity; (ii) explicit comparison of quadratic energies showing that the ray-monotone plan is the unique C₂-minimizer for every (μ, ν_n); (iii) verification of weak convergence γ^sel(μ, ν_n) ⇀ γ^hom by testing against continuous functions; and (iv) geometric inspection of the limit supports together with a quadratic-energy comparison proving that γ^hom is not ray-monotone (hence not the C₂-selector) for (μ, ν). These additions will make every asserted property fully checkable without changing the results. revision: yes

  2. Referee: §4 (narrow Kuratowski limit of O(μ, ν_n)): the identification of the limit set must be shown to be compatible with the weak convergence of the selected plans; if the Kuratowski limit contains plans other than γ^hom, the argument that γ^hom is the only possible weak limit point of the sequence γ^sel(μ, ν_n) needs an additional tightness or uniqueness argument.

    Authors: We appreciate the referee’s observation on compatibility. The manuscript identifies the narrow Kuratowski limit of O(μ, ν_n), which properly contains γ^hom. To address the concern we will insert a short paragraph in §4 establishing tightness of {γ^sel(μ, ν_n)} (uniformly bounded supports and W₁-optimality yield Prokhorov tightness) and showing that any weak limit point must minimize the secondary C₂ energy in the limit; the latter property singles out γ^hom among all elements of the Kuratowski limit. This supplies the missing uniqueness argument while leaving the Kuratowski identification unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central contribution is an explicit construction of a fixed absolutely continuous μ and a sequence of absolutely continuous ν_n ⇀ ν such that the C₂-selected plans γ^sel(μ,ν_n) converge weakly to a W₁-optimal plan γ^hom for the limit pair that differs from γ^sel(μ,ν). This is a direct counterexample to stability of the ray-monotone selector, not a derivation that reduces by construction to its own inputs, fitted parameters renamed as predictions, or a self-citation chain. The abstract and claimed results (Kuratowski limit of O(μ,ν_n), constrained Γ-limit of secondary energies, non-commutation for c_ε) follow from the geometry of the constructed supports and densities under standard absolute-continuity hypotheses; no load-bearing step equates a claimed output to the construction itself or to an unverified self-citation. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of specific absolutely continuous measures satisfying optimality and convergence properties under the distance cost, relying on standard results from optimal transport theory such as existence of optimal plans and properties of weak convergence.

axioms (2)
  • standard math Standard properties of weak convergence of measures and optimal transport plans in Euclidean space under the distance cost.
    Invoked implicitly throughout the statements of convergence of plans and marginals.
  • domain assumption Absolute continuity of source and target measures ensures the ray-monotone plan is obtained via secondary variational selection by quadratic energy.
    Stated as the classical hypothesis under which the selector is defined.

pith-pipeline@v0.9.0 · 5540 in / 1355 out tokens · 91928 ms · 2026-05-10T10:07:43.866789+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    Existence and stability results in theL1 theory of optimal trans- portation

    Luigi Ambrosio and Aldo Pratelli. Existence and stability results in theL1 theory of optimal trans- portation. InOptimal Transportation and Applications, volume 1813 ofLecture Notes in Mathematics, pages 123–160. Springer, Berlin, 2003. DOI:https://doi.org/10.1007/978-3-540-44857-0_5

    work page doi:10.1007/978-3-540-44857-0_5 2003

  2. [2]

    [3]Altschuler, J

    Shayan Aryan and Promit Ghosal. Entropic selection principle for Monge’s optimal transport. https://arxiv.org/abs/2502.16370, 2025. Preprint, arXiv:2502.16370 [math.PR]

    work page arXiv 2025

  3. [3]

    A proof of Monge problem inRn by stability.Rend

    Laura Caravenna. A proof of Monge problem inRn by stability.Rend. Istit. Mat. Univ. Trieste, 43:31–51, 2011

    work page 2011

  4. [4]

    The Monge problem for strictly convex norms inRd.J

    Thierry Champion and Luigi De Pascale. The Monge problem for strictly convex norms inRd.J. Eur. Math. Soc., 12(6):1355–1369, 2010. DOI:https://doi.org/10.4171/JEMS/234

    work page doi:10.4171/jems/234 2010

  5. [5]

    The Monge problem inRd.Duke Math

    Thierry Champion and Luigi De Pascale. The Monge problem inRd.Duke Math. J., 157(3):551–572,

    work page

  6. [6]

    DOI:https://doi.org/10.1215/00127094-1272939

    work page doi:10.1215/00127094-1272939

  7. [7]

    Existence and uniqueness of optimal transport maps obtained by the secondary variational method.Pacific J

    Peng Chen, Hongrui Liu, and Xiaoping Yang. Existence and uniqueness of optimal transport maps obtained by the secondary variational method.Pacific J. Math., 312:75–102, 2021. DOI: https://doi.org/10.2140/pjm.2021.312.75. 23

    work page doi:10.2140/pjm.2021.312.75 2021

  8. [8]

    Quantitative stability of optimal transport maps under variations of the target measure.Duke Math

    Adrien Delalande and Quentin Mérigot. Quantitative stability of optimal transport maps under variations of the target measure.Duke Math. J., 172(17):3321–3357, 2023. DOI:https://doi.org/ 10.1215/00127094-2022-0106

    work page doi:10.1215/00127094-2022-0106 2023

  9. [9]

    The entropic regularization of the Monge problem on the real line.SIAM J

    Simone Di Marino and Julien Louet. The entropic regularization of the Monge problem on the real line.SIAM J. Math. Anal., 50(4):3451–3477, 2018. DOI:https://doi.org/10.1137/17M1123523

    work page doi:10.1137/17m1123523 2018

  10. [10]

    Unstable optimal transport maps.https://arxiv.org/abs/2510.13265, 2025

    Clément Letrouit. Unstable optimal transport maps.https://arxiv.org/abs/2510.13265, 2025. Preprint, arXiv:2510.13265 [math.OC]

    work page arXiv 2025

  11. [11]

    Entropic selection for optimal transport on the line with distance cost

    Antoine Ley. Entropic selection for optimal transport on the line with distance cost. https: //arxiv.org/abs/2512.05282, 2025. Preprint, arXiv:2512.05282 [math.OC]

    work page arXiv 2025

  12. [12]

    Absolute continuity and summability of transport densities: simpler proofs and new estimates.Calc

    Filippo Santambrogio. Absolute continuity and summability of transport densities: simpler proofs and new estimates.Calc. Var. Partial Differential Equations, 36:343–354, 2009. DOI: https: //doi.org/10.1007/s00526-009-0231-8

    work page doi:10.1007/s00526-009-0231-8 2009

  13. [13]

    Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling , year =

    Filippo Santambrogio.Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, volume 87 ofProgress in Nonlinear Differential Equations and Their Applications. Birkhäuser/Springer, Cham, 2015. DOI:https://doi.org/10.1007/978-3-319-20828-2. 24

    work page doi:10.1007/978-3-319-20828-2 2015