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Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance

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Pith's one-line read Under Hölder regularity, optimal transport maps between non-degenerate densities on uniformly convex domains are Lipschitz stable in L² with respect to the 2-Monge-Kantorovich distance.

desk verdict Linearization of the divergence-form Monge-Ampère equation yields both the stability estimates and an explicit second-variation formula for the quadratic MK distance. read the letter →

arxiv 2605.24232 v1 pith:ISPUVPR5 submitted 2026-05-22 math.AP

F.-U. Caja-Lopez , Matias G. Delgadino , Jun Kitagawa This is my paper · ORCID
classification math.AP
keywords optimaltransportMonge-AmpèreequationstabilityestimatesBrenierpotentialsMonge-Kantorovichdistancelinearizationsecondvariationconvexdomains
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives quantitative stability estimates for optimal transport maps with quadratic cost. When source and target densities are Hölder continuous and non-degenerate on uniformly convex domains, the maps satisfy a Lipschitz bound in L² controlled by the 2-Monge-Kantorovich distance between the densities. They also obtain C^{1,α} stability controlled by L^p distances. Without the Hölder assumption the Brenier potentials are still Lipschitz stable in L² with respect to the L² distance of the densities. The key step is a characterization of the linear response of the Brenier potential obtained by linearizing the Monge-Ampère equation, which additionally yields an explicit second variation formula for the Monge-Kantorovich distance.

What carries the argument

The linearized Monge-Ampère equation in divergence form, used to characterize the first variation of the Brenier potential along interpolations of the densities.

What would settle it

Finding a counterexample consisting of non-degenerate Hölder densities on a uniformly convex domain where the L² difference of the optimal maps is not bounded by a constant times the 2-Monge-Kantorovich distance between the densities.

Watch

Extended reading notes

Core claim

Under the stated assumptions the linear response of the Brenier potential to perturbations of the source and target densities is given by the solution of the linearized Monge-Ampère equation in divergence form. This response implies the Lipschitz stability estimates for the maps and potentials, and it produces the explicit formula for the second variation of the quadratic Monge-Kantorovich distance.

Load-bearing premise

Densities are non-degenerate on uniformly convex domains, with Hölder continuity needed for the map stability claims.

Editorial extensions

If this is right

  • The optimal transport map depends Lipschitz continuously on the pair of densities when measured in the 2-Monge-Kantorovich metric and L² norm.
  • C^{1,α} regularity of the map is controlled by L^p distances of the densities under Hölder assumptions.
  • Brenier potentials remain stable in L² even if the densities lose continuity.
  • The second variation of the 2-Monge-Kantorovich distance admits an explicit integral expression derived from the linearization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • This linearization technique could be applied to derive stability results for other cost functions in optimal transport.
  • The stability estimates suggest quantitative convergence rates for approximations of transport maps when densities are perturbed by discretization error.
  • Connections to the geometry of the Wasserstein space may follow from the second variation formula.
  • Testing the formula on explicit examples such as uniform densities on balls would verify the second variation explicitly.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper establishes quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under Hölder regularity assumptions on the densities, it proves Lipschitz L² stability estimates for the maps in terms of the 2-Monge-Kantorovich distance between source and target densities, and C^{1,α} stability in terms of L^p distances. Removing the continuity assumption yields a Lipschitz L² stability estimate for the Brenier potentials in terms of the L² distance between the densities. The proofs rely on a precise characterization of the linear response of the Brenier potential obtained by linearizing the Monge-Ampère equation in divergence form along smooth interpolations; as an application, an explicit formula for the second variation of the quadratic Monge-Kantorovich distance is derived.

Significance. If the central claims hold, the results supply useful quantitative stability controls for optimal transport maps and potentials under standard structural assumptions (uniform convexity of domains and non-degeneracy of densities). The linearization approach to the Monge-Ampère equation and the explicit second-variation formula constitute concrete contributions that can be applied in stability analysis and related PDE problems. The distinction between map stability under Hölder regularity and potential stability without continuity is clearly delineated.

major comments (2)
  1. [§3] §3 (linearization step): the derivation of the linearized Monge-Ampère equation in divergence form assumes sufficient smoothness of the interpolation to justify differentiation under the integral; the manuscript should explicitly verify that the error terms arising from this differentiation remain controlled by the 2-MK distance under the stated Hölder assumptions.
  2. [Theorem 1.2] Theorem 1.2 (map stability): the Lipschitz constant in the L² estimate appears to depend on the uniform convexity constants and the non-degeneracy bounds; the proof should confirm that this dependence is explicit and does not deteriorate when the Hölder exponent α approaches zero.
minor comments (2)
  1. [Abstract] The abstract states the results for 'pairs of source and target densities' but the precise interpolation path between densities is only defined later; a brief sentence in the abstract or introduction clarifying the interpolation would improve readability.
  2. [Introduction] Notation for the 2-Monge-Kantorovich distance is introduced without an explicit formula in the opening paragraph; adding the standard definition W_2^2(μ,ν) = inf ∫ |x-y|^2 dγ would help readers unfamiliar with the notation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we address the two major comments point by point, indicating the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (linearization step): the derivation of the linearized Monge-Ampère equation in divergence form assumes sufficient smoothness of the interpolation to justify differentiation under the integral; the manuscript should explicitly verify that the error terms arising from this differentiation remain controlled by the 2-MK distance under the stated Hölder assumptions.

    Authors: We agree that the passage from the smooth interpolation to the linearized equation requires explicit control of the error terms arising from differentiation under the integral. In the revised manuscript we will add a short lemma in §3 that bounds these remainder terms in terms of the 2-MK distance, using only the Hölder regularity of the densities and the uniform convexity of the domains. The argument proceeds by approximating the given Hölder densities by smooth ones whose 2-MK distance is controlled, applying the smooth linearization, and passing to the limit with the error estimate. revision: yes

  2. Referee: [Theorem 1.2] Theorem 1.2 (map stability): the Lipschitz constant in the L² estimate appears to depend on the uniform convexity constants and the non-degeneracy bounds; the proof should confirm that this dependence is explicit and does not deteriorate when the Hölder exponent α approaches zero.

    Authors: The Lipschitz constant in Theorem 1.2 is explicit and depends on the uniform convexity constants, the non-degeneracy bounds, and the Hölder exponent α. As α → 0 the constant necessarily deteriorates; this is consistent with the paper’s separation of results, since map stability in L² requires positive Hölder regularity while the continuous case yields only potential stability. We will insert a remark immediately after the statement of Theorem 1.2 that records the explicit dependence on all parameters and notes the blow-up as α → 0. No change to the theorem statement itself is required. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The derivation relies on linearizing the Monge-Ampère equation in divergence form to characterize the linear response of the Brenier potential, followed by standard estimates under Hölder or continuity assumptions on non-degenerate densities. No quoted step reduces a claimed stability estimate or second-variation formula to a fitted parameter, self-defined quantity, or load-bearing self-citation; the argument is a direct PDE analysis whose inputs (uniform convexity, non-degeneracy) are independent of the output estimates.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The work rests on standard domain assumptions of optimal transport theory; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

assumptions (2)
  • domain assumption Densities are non-degenerate on uniformly convex domains
    Required setting for the stability statements (abstract opening sentence).
  • domain assumption Hölder regularity of densities for the C^{1,α} map estimates
    Explicitly assumed to obtain the stronger stability result.

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Pith. "Pith review of Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance." pith.science (2026). https://pith.science/paper/ISPUVPR5

@misc{pith2026260524232,
  author       = {Pith},
  title        = {Pith review of: Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISPUVPR5}},
  note         = {Machine review of arXiv:2605.24232}
}
abstract

We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under H\"older regularity assumptions, we prove Lipschitz $L^2$ (respectively $C^{1,\alpha}$) stability estimates for optimal transport maps in terms of the 2-Monge-Kantorovich distance (respectively $L^{p}$ distances) between pairs of source and target densities. When the continuity assumption is removed, we obtain a Lipschitz $L^2$ stability estimate for the Brenier potentials in terms of the $L^2$ distance between the source and target densities. The proofs rely on a precise characterization of the linear response of the Brenier potential along smooth interpolations of the data, obtained by linearizing the Monge-Amp\`ere equation in divergence form. As a further application of this approach, we derive an explicit formula for the second variation of the quadratic Monge-Kantorovich distance.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. No-gap second-order conditions for optimization problems involving transport distances

    math.OC 2026-07 accept novelty 6.0 of 10

    No-gap second-order conditions for transport-regularized measure optimization are equivalent to quadratic growth once the Kantorovich potential satisfies a quadratic-growth regularity assumption.

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