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Global Regularity Estimates for Optimal Transport via Entropic Regularisation

Nathael Gozlan, Maxime Sylvestre

TL;DR

This work develops a flexible entropic-regularization framework to obtain global regularity estimates for Brenier maps in quadratic optimal transport. By coupling entropic Legendre transforms with Prékopa–Leindler inequalities, the authors derive ε-independent modulus bounds for Kantorovich potentials, enabling anisotropic and directional regularity results beyond classical Euclidean settings. They establish Hölder and Lipschitz-type control under various convexity/smoothness hypotheses, provide non-Euclidean norm analogues, and extend Lipschitz-growth and concentration results for perturbations of Gaussian measures. The framework also yields growth bounds and a divergence-control result for the Brenier map, with implications for functional inequalities and large deviations. Overall, the paper advances global regularity theory in OT by unifying entropic, variational, and convex-analytic methods with broad applicability to singular measures and anisotropic geometries.

Abstract

We develop a general approach to prove global regularity estimates for quadratic optimal transport using the entropic regularisation of the problem and the Prekopa-Leindler inequality.

Global Regularity Estimates for Optimal Transport via Entropic Regularisation

TL;DR

This work develops a flexible entropic-regularization framework to obtain global regularity estimates for Brenier maps in quadratic optimal transport. By coupling entropic Legendre transforms with Prékopa–Leindler inequalities, the authors derive ε-independent modulus bounds for Kantorovich potentials, enabling anisotropic and directional regularity results beyond classical Euclidean settings. They establish Hölder and Lipschitz-type control under various convexity/smoothness hypotheses, provide non-Euclidean norm analogues, and extend Lipschitz-growth and concentration results for perturbations of Gaussian measures. The framework also yields growth bounds and a divergence-control result for the Brenier map, with implications for functional inequalities and large deviations. Overall, the paper advances global regularity theory in OT by unifying entropic, variational, and convex-analytic methods with broad applicability to singular measures and anisotropic geometries.

Abstract

We develop a general approach to prove global regularity estimates for quadratic optimal transport using the entropic regularisation of the problem and the Prekopa-Leindler inequality.
Paper Structure (27 sections, 50 theorems, 303 equations)

This paper contains 27 sections, 50 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. Modulus of convexity and modulus of smoothness
  3. Modulus of convexity/smoothness
  4. Directional moduli of convexity and smoothness
  5. Examples of $R$-convex and $S$-smooth functions
  6. Quadratic functions.
  7. Functions with a bounded hessian.
  8. Functions with a uniformly continuous gradient.
  9. Radial functions.
  10. Entropic Legendre transform
  11. $\psi$ $\rho$-convex implies $\mathcal{L}_{\epsilon,\mathcal{H}^n}(\psi)$ $\rho^\ast$-smooth
  12. $\psi$ $\sigma$-smooth implies $\mathcal{L}_{\epsilon,\mathcal{H}^n}(\psi)$ $\sigma^\ast$-convex
  13. A remark on $\rho$-convexity along the HJB flow
  14. A remark on the Cramér transform
  15. Application to optimal transport
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $\mu(dx) = e^{-V(x)}dx$, $\nu(dy) = e^{-W(y)} dy$ be two probability measures on ${\mathbb R}^n$ such that $\mathrm{dom} V = {\mathbb R}^n$ and $\mathrm{dom}W$ is convex with nonempty interior. Further assume that $V,W$ are twice continuously differentiable on the interior of their domains and s with $\alpha_V,\beta_W >0$. Then the optimal transport map for the quadratic transport problem from

Theorems & Definitions (63)

  • Theorem 1.1: Caffarelli Caffarelli2000Caf02
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proposition 2.9
  • ...and 53 more