A uniform rate of convergence for the entropic potentials in the quadratic Euclidean setting

TL;DR

This work establishes quantitative uniform convergence rates for entropic potentials in the quadratic Euclidean optimal transport setting. Under convexity-type assumptions on the source and target measures, the authors prove that the entropic Brenier potentials $\varphi_\varepsilon$ and their gradients $\nabla\varphi_\varepsilon$ converge to the Brenier potential $\varphi_0$ and gradient $\nabla\varphi_0$ at rates $O(\varepsilon^{1/(d+4)})$ on compact sets, with a corresponding $O(\varepsilon^{1/(d+4)}+\varepsilon)$ rate for the potentials themselves when a normalization is imposed. The proof leverages convexity of the entropic potentials, a Gagliardo–Nirenberg inequality framework, and second-order estimates to relate sup-norm gradient convergence to higher-derivative controls, complemented by Evans-type bounds to transfer gradient estimates to potential convergence. A Gaussian case is worked out explicitly, yielding a dimension-free $O(\varepsilon)$ bound on compact sets with a precise matrix expression, illustrating the theoretical rates with a concrete calculation. These results meaningfully quantify the rate at which entropic regularization approximates classical OT in the continuous setting and have implications for numerical schemes based on entropic regularization, such as Sinkhorn-type methods.

Abstract

We bound the rate of uniform convergence in compact sets for both entropic potentials and their gradients towards the Brenier potential and its gradient, respectively. Both results hold in the quadratic Euclidean setting for absolutely continuous measures satisfying some convexity assumptions.

Theorems & Definitions (18)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Theorem 2.2: Caffarelli
• Theorem 2.3
• Remark 2.1
• Remark 2.2
• Remark 2.3