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arxiv: 2604.22670 · v1 · submitted 2026-04-24 · 🧮 math.OC · math.PR

Curvature of optimal transport with respect to the cost and applications to inverse optimal transport

Gabriel Peyr\'e , Clarice Poon , Oscar Tron This is my paper

Pith reviewed 2026-05-08 10:56 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords optimal transportinverse optimal transportcurvatureidentifiabilitystabilityentropic regularizationMahalanobis costscontinuous setting
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The pith

Smooth positive densities make the optimal transport functional strictly curved with respect to the ground cost.

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

The paper aims to resolve the ill-posedness of recovering the ground cost from an observed optimal transport plan, a problem that is intrinsically degenerate in discrete settings. By working in the continuous regime with smooth positive source and target densities, the authors derive the second variation of the optimal transport cost functional with respect to cost perturbations in appropriate Hölder spaces. They prove this variation is positive modulo the natural invariances of optimal transport, establishing a strict local curvature that supports local identifiability and stability of the inverse problem. This yields concrete recovery results for parametrized families such as Mahalanobis costs, together with statistical bounds under entropic smoothing.

Core claim

Assuming smooth positive densities for the source and target measures, we characterize the second variation of the optimal transport functional with respect to the ground cost in Hölder spaces. In particular, we show that it is non-degenerate modulo the natural transport invariances, yielding a strict curvature property that is absent in discrete transport. As a consequence, we obtain local identifiability and stability results for inverse optimal transport. For the structured family of bilinear costs, the ground cost can be uniquely recovered up to the intrinsic invariances from a single optimal coupling under a natural spanning condition.

What carries the argument

The second variation of the optimal transport functional with respect to the ground cost, characterized in Hölder spaces and proven non-degenerate modulo invariances.

Load-bearing premise

The source and target measures must have smooth positive densities, without which the second variation may degenerate and the inverse problem stays ill-posed.

What would settle it

Finding a pair of smooth positive densities where the second variation operator admits a kernel larger than the natural invariances, or where the inverse map fails to be locally unique, would disprove the claimed non-degeneracy.

Figures

Figures reproduced from arXiv: 2604.22670 by Clarice Poon, Gabriel Peyr\'e, Oscar Tron.

Figure 1
Figure 1. Figure 1: L n 0 on the diagonal for Gaussian to Gaussian (blue) and Gaussian to Perturbed Gaussian (red) iOT (a) n = 5 samples (b) n = 10 samples (c) n = 20 samples (d) n = 30 samples view at source ↗
Figure 2
Figure 2. Figure 2: L n 0 on the diagonal for the Gaussian to Perturbed Gaussian setting. Numerical illustration In view at source ↗
Figure 3
Figure 3. Figure 3: L n 0 on the diagonal for Gaussian/Gaussian (blue) and Gaussian/annulus (red) iOT view at source ↗
Figure 4
Figure 4. Figure 4: displays the transport map TA for the previous Gaussian/annulus case as arrows acting on n = 1000 samples for different matrices A = diag(u, 1 − u) for u ∈ {0.5, 0.9, 0.1}. The map for u = 0.5 transports the middle of the Gaussian quite uniformly to the annulus, whereas for u = 0.9 (resp. u = 0.1) the transport has a horizontal (resp. vertical) separation. This illustrates, in that case, the dependency of … view at source ↗
Figure 5
Figure 5. Figure 5: Jε on [0, 1]2 with R = ∥∥∗ Conclusion This work establishes a sharp contrast between inverse OT regimes: from a single coupling, the problem is intrinsically degenerate in discrete/polyhedral or highly symmetric settings, while smooth continuous marginals can restore local well-posedness through transport curvature. Theorem 1 characterizes the second-order geometry of OT with respect to the cost function. … view at source ↗
read the original abstract

We study the inverse optimal transport problem of recovering the ground cost from an optimal transport plan. In discrete settings, this problem reduces to inverse linear programming and is intrinsically ill-posed, exhibiting non-identifiability and flat directions. We show that in the continuous setting, the regularity of the marginals fundamentally alters the structure of the inverse problem. Assuming smooth positive densities for the source and target measures, we characterize the second variation of the optimal transport functional with respect to the ground cost in H\"older spaces. In particular, we show that it is non-degenerate modulo the natural transport invariances, yielding a strict curvature property that is absent in discrete transport. As a consequence, we obtain local identifiability and stability results for inverse optimal transport. For the structured family of bilinear costs (i.e. Mahalanobis parametrizations), the ground cost can be uniquely recovered--up to the intrinsic invariances--from a single optimal coupling under a natural spanning condition. We further show that this identifiability property is generic under arbitrarily small perturbations of the marginals, while settings where the optimal transport map is affine (for instance Gaussian or elliptical marginals) remain degenerate. Finally, we establish precise bounds on the bias and statistical variance of inverse optimal transport under entropic regularization. These results reveal a structural parallel between forward and inverse optimal transport: regularity of the marginals ensures smooth optimal maps in the forward problem, while non-degeneracy of the induced transport plan yields curvature and local invertibility in the inverse problem.

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 studies the inverse optimal transport (IOT) problem of recovering the ground cost from an observed optimal transport plan. Unlike the discrete case, where IOT reduces to ill-posed inverse linear programming with non-identifiability, the paper shows that smooth positive densities on the source and target measures induce a non-degenerate second variation of the OT functional with respect to the cost (in Hölder spaces). This yields a strict curvature property modulo the natural invariances (additive functions f(x) + g(y)), implying local identifiability and stability for IOT. For the structured family of bilinear (Mahalanobis) costs, unique recovery up to invariances holds from a single coupling under a spanning condition. Identifiability is generic under small marginal perturbations, but remains degenerate when the OT map is affine (e.g., Gaussian or elliptical marginals). Precise bias and variance bounds are also derived for entropic regularization of IOT.

Significance. If the central claims hold, the work establishes a fundamental structural distinction between discrete and continuous IOT, providing a rigorous basis for well-posedness and local invertibility in the continuous setting. Key strengths include the explicit linearization of the optimality conditions, the resulting integro-differential operator, and the proof that its kernel coincides exactly with the transport invariances on the quotient space; these enable the curvature lower bound and local stability. The generic perturbation result and the statistical bounds under entropic regularization further strengthen applicability. The parallel drawn between regularity in forward OT (smooth maps) and non-degeneracy in inverse OT is insightful and could influence cost-recovery problems in economics, statistics, and machine learning.

minor comments (3)
  1. [§3] §3 (second-variation characterization): the passage from the linearized Monge-Ampère equation to the Hölder-space curvature bound would benefit from an explicit statement of the constant in the lower bound (currently implicit in the non-degeneracy argument).
  2. [Theorem 4.2] Theorem 4.2 (bilinear-cost identifiability): the spanning condition on the support of the coupling is stated clearly, but a brief remark on how it fails for affine maps (as in the Gaussian case of §5) would improve readability.
  3. [§6] The entropic-regularization bounds in §6 are precise, yet the dependence of the variance term on the regularization parameter ε could be highlighted in a single displayed inequality for quick reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work on inverse optimal transport and for recommending minor revision. The provided summary correctly identifies the key distinction between discrete and continuous settings, the role of smooth marginals in inducing non-degenerate curvature, and the resulting local identifiability and stability results. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes its central curvature and identifiability claims through direct variational analysis: it linearizes the optimality conditions of the OT problem to derive an integro-differential operator on the second variation of the functional with respect to the cost (in Hölder spaces), then proves that the kernel of this operator coincides exactly with the natural invariances (additive functions f(x) + g(y)) when the marginals are smooth and positive. Local invertibility and stability follow from the resulting strict positivity on the quotient space. This derivation is self-contained within the stated assumptions and does not reduce any prediction or uniqueness statement to a fitted parameter, self-definition, or load-bearing self-citation chain. Minor references to prior OT regularity results appear but are not invoked as external uniqueness theorems that close the argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption of smooth positive densities and on the existence of natural transport invariances; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Smooth positive densities for the source and target measures
    Stated as the condition that fundamentally alters the structure of the inverse problem and yields non-degeneracy.

pith-pipeline@v0.9.0 · 5582 in / 1181 out tokens · 31913 ms · 2026-05-08T10:56:40.106974+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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    log(n)s n !! The first step is to bound the entropic cost withε= 1 andc(x, y) = 1 2 |x−y| 2 by a Rademacher complexity. The following proposition comes from [Mena and Niles-Weed, 2019]. Proposition A.6.Assume thatαandβareσ 2-sub Gaussian. Let’s write the random sub Gaussian constant˜σ= inf{σ >0,s.t.α n, α, βn andβareσ 2-sub Gaussian}. Then withs=⌈d/2⌉+ 1:...

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    Then for all invertible matrixA∈R d×d 0≤I(A)≤ sup∥A∥=1 I(A) ∥A∥ wheresup ∥A∥=1 I(A)<∞ Proof.From lemma B.1 we know thatIis continuous and therefore bounded on the unit ball. Moreover,A7→T A 0 is invariant by positive translation:T λA 0 =T A 0 providedλ >0, because positively collinear matrices generate the same classic OT plan. Thereforeρ λA t =ρ A t and ...

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