2026/08/01 by Alberto González-Sanz, Gilles Mordant, Shunan Sheng
Mathematics · #math.ST #stat.TH #msc:62G35 #msc:62G30
arxiv created 2026/08/01 · arxiv updated 2026/08/04
Optimal transport potentials are fundamental objects in statistics, economics, and machine learning: their gradients generate optimal transport maps, while the potentials themselves act as location-dependent dual prices and sensitivity variables. We study the estimation of the quadratic optimal transport potential when a fixed absolutely continuous reference distribution μ is transported to an unknown distribution ν, accessed to via its empirical measure. Our main ingredient is a stability inequality that controls the L1(μ) distance, modulo additive constants, between a strongly convex potential φ and a convex potential \widetildeφ by a weak dual norm of (∇\widetildeφ)_#μ-(∇φ)_#μ, together with a second-order Wasserstein remainder of logarithmic type. This separation between the leading empirical-process term and the Wasserstein remainder yields faster convergence for potentials than for the corresponding transport maps. Under smoothness and uniform convexity assumptions, the exact semidiscrete Brenier potential converges in L1(μ) at rate n-1/2 for d≤3, at rate n-1/2(log n)5/2 for d=4, and at rate n-2/d(log n)(d+2)/d for d≥5. The polynomial exponents are sharp. In dimensions d≤3, we further establish a nondegenerate function-space central limit theorem and prove consistency of the nonparametric bootstrap. These results yield joint root-n inference for every fixed finite collection of normalization-invariant weighted contrasts of the potential, including regional shadow premia in reference-based risk problems. Finally, we prove matching upper and lower bounds of order εlog(1/ε) for the normalization-invariant sum of the entropic dual potentials.