Quantitative Stability, Coercivity and Uniqueness of Optimal Transport Plans

1.1. Contributions.

• The regularity assumption bounds the mass of every radius-η ball by Mη^(d−1+θ), with θ>0, and is presented as a quantitative form of the Brenier-McCann hypothesis.The foundational qualitative result cited in the introduction gives uniqueness and a map-induced optimal plan when the source vanishes on sets of Hausdorff dimension d−1.Source: §1.1 · p. 2
• Under A(M, θ), optimal plans are quantitatively stable when both marginals are perturbed, with the bound sharp in exponent.The introduction also states fixed-source map stability results; with a fixed source, map stability implies plan stability with the same constant and exponent.Source: §1.1 · p. 2
• A plan with similar marginals and transport cost to the optimum is quantitatively close to the optimal plan when the source satisfies A(M, θ).The paper also gives coercivity results for problems with a marginal close to a regular measure and for maps built from subgradients of nearly optimal dual potentials.Source: §1.1 · pp. 2–3
• The Wasserstein diameter of optimal plans is bounded by a marginal’s distance to a measure satisfying A(M, θ), quantifying near-uniqueness regardless of the target.The paper presents this as a quantitative counterpart to the Brenier-McCann theorem; the stated bound is sharp in exponent.Source: §1.1 · p. 3

1.2. Existing results.

• Prior stability work assumes either a regular transport map or potential, or a regular marginal; this paper’s stability results belong to the latter category.The introduction notes that map or potential regularity can be difficult to verify and maps may fail to be continuous, even between regular densities.Source: §1.2 · p. 3
• The cited primal uniqueness literature is qualitative, while earlier coercivity results use regular-potential assumptions or fixed marginals.The paper describes its regularity condition as a quantitative strengthening of the Brenier-McCann hypothesis.Source: §1.2 · p. 4

1.3. Idea of the proofs.

• The proofs couple plans optimally for the first marginal and bound Fenchel–Young slack using marginal distances, plus the cost difference for coercivity.The paper’s quantitative regularity estimate for convex functions extends the cited result beyond Lebesgue measure and bounded densities.Source: §1.3 · p. 5

2. Quantitative regularity: coupling transport plans

• The paper normalizes domains by translation when possible and uses compact convex sets without requiring measures to have full support.The associated radii and diameters characterize the domains.Source: §2 · p. 5

2.1. An integral bound.

• Theorem 2.2 gives an integral bound for gradients of convex functions when the reference measure satisfies A(M, θ).Its bounds extend earlier work to q = 1 and measures satisfying A(M, θ).Source: §2.1 · p. 6

2.2. The Fenchel gap.

• The Fenchel–Young remainder measures how far a pair is from equality, which is equivalent to belonging to the graph of the convex function’s subgradient.The section’s lemma bounds distance to subgradients using this gap.Source: §2.2 · p. 7

2.3. Coupling transport plans.

• Couplings of plans are built from a coupling of their first marginals and conditional plan distributions, giving a competitor for their Wasserstein distance.Choosing the marginal coupling optimally controls one term; the other is the main quantity to bound.Source: §2.3 · p. 8

3.1. Quantitative uniqueness.

• When one marginal is close to a measure satisfying A(M, θ), the Wasserstein diameter of the optimal plans is quantitatively controlled.The theorem assumes compact convex domains and a reference measure satisfying A(M, θ).Source: §3.1 · p. 9, §3 · p. 9

3.2. Plans on the same subgradient with different marginals.

• The different-marginal comparison generalizes [14], improves its exponent, and gives a Wq–Wr bound rather than a W2–W2 bound.The exponent is shown sharp in Proposition 5.1, and the theorem also gives an alternative proof of plan stability.Source: §3.2 · p. 10
• The same-subgradient plan bound compares a regular-source plan with another plan on that subgradient whose first marginal may differ.The theorem applies for q ≥ 1 and r ∈ [1, ∞].Source: §3.2 · p. 10

4.1. Coercivity for plans with arbitrary marginals.

• A plan with arbitrary marginals is quantitatively close to the optimal plan when its marginals and transport cost are close to the reference problem’s.The reference source measure satisfies A(M, θ).Source: §4.1 · p. 12

4.2. Approximate coercivity with an almost regular source measure.

• Coercivity also holds with an almost regular source, with an error quantified by its Wasserstein distance to a measure satisfying A(M, θ).For r other than 1, this result has a sharper error exponent than the preceding coercivity theorem combined with stability.Source: §4.2 · p. 13

4.3. Coercivity of maps with respect to dual suboptimality.

• The map coercivity theorem bounds deviation from the optimal map using dual suboptimality when the source satisfies A(M, θ).It extends [33, Theorem 3.2] to general q and θ.Source: §4.3 · p. 14

4.4. Stability of optimal maps.

• Bi-marginal map stability compares gradients of Brenier potentials in Lq(ρ0), even though a Brenier map need not exist for the perturbed pair.The theorem applies to Brenier potentials satisfying (2.1); their values outside the perturbed source support have no canonical choice.Source: §4.4 · p. 15
• The map-stability bound gives an approximator in Lq(ρ0) when a Brenier potential is available for the perturbed marginals.Its gradient sends samples from ρ0, on average, near the image under the true optimal map.Source: §4.4 · p. 15

4.5. Stability of optimal plans.

• Optimal plans are quantitatively stable under perturbations of both marginals when the reference source satisfies A(M, θ).The result applies to any optimal plan for the perturbed marginals.Source: §4.5 · p. 16

5.1. Sharpness for uniqueness.

• The cited passages conclude that Theorems 3.1 and 3.3 are optimal in exponent for all q, r, and θ; they also state Theorem 3.1 cannot generally hold with a better exponent.Source: §5.1 · pp. 17–18

5.2. Sharpness for stability of maps and plans.

• A mass-based pigeonhole argument lower-bounds plan distances when the plans’ supports are not cleanly separated.The second example establishes sharpness for map and plan stability; the resulting exponents in Theorems 4.6 and 4.8 cannot be improved.Source: §5.1 · p. 18, §5.2 · pp. 18, 21

5.3. Sharpness for coercivity.

• Fixed-marginal examples show that the exponents in Theorems 4.2 and 4.4 are sharp, including the exponent in terms of the suboptimality deficit.The example adapts the stability construction and uses equal marginals.Source: §5.3 · pp. 21–22

• The appendix presents a second proof of Theorem 4.8 and proves strong convexity of the Kantorovich functional, generalizing to the paper’s exponents q and θ.Source: Appendix A · p. 22

A.1. An alternative proof of Theorem 4.8: bi-marginal plan stability.

• Coupling graph plans through a common source and applying Theorem 4.6 recovers Theorem 4.8’s exponent, possibly with a different constant.Source: §A.1 · p. 22

A.2. Strong convexity of the Kantorovich functional via the Jensen divergence.

• The section measures strong convexity of Kρ using the Jensen divergence Jρ.Kρ is convex and 1-Lipschitz in the uniform norm; under A(M, θ), it is everywhere Gâteaux differentiable.Source: §A.2 · p. 22
• The theorem generalizes [33, Theorem 2.2], which is recovered for θ = 1 and q = 2.Source: §A.2 · p. 22
• The stated theorem assumes compact convex X and Y in R^d, d ≥ 2, and a probability measure ρ on X satisfying A(M, θ).It applies to all ψ0, ψ1 in C0(Y).Source: §A.2 · p. 22