标题： 一般分布之间最优传输映射估计的统计收敛速率 显示英文标题

标题： Statistical Convergence Rates of Optimal Transport Map Estimation between General Distributions

摘要： 本文研究了最优传输（OT）映射估计器的收敛速率，这是一个在统计学、机器学习和各种科学领域日益受到关注的主题。 尽管最近取得了进展，现有的结果依赖于在实践中非常限制性的正则性假设，这些假设比布伦纳定理中的假设要严格得多，包括概率支撑的紧性和凸性以及OT映射的双利普希茨性质。 我们旨在拓宽OT映射估计的范围，并填补理论与实践之间的这一差距。 在对布伦纳势强凸性假设的基础上，我们首先建立了原始插值估计器的非渐近收敛速率，而无需对概率测度施加限制性假设。 此外，我们引入了一种筛子插值估计器，并在其不依赖于布伦纳势的强凸性假设的情况下建立了其收敛速率，从而能够处理诸如正态分布或t分布的秩函数等广泛应用的情况。 我们还建立了新的泊松型不等式，在概率密度的局部有界性和支撑的温和拓扑条件下的充分条件下得到了证明，这些新不等式使我们能够为多恩斯克函数类实现更快的收敛速率。 此外，我们开发了可扩展的算法，利用神经网络高效求解OT映射估计，并通过数值实验展示了其有效性和鲁棒性。 显示英文摘要

摘要： This paper studies the convergence rates of optimal transport (OT) map estimators, a topic of growing interest in statistics, machine learning, and various scientific fields. Despite recent advancements, existing results rely on regularity assumptions that are very restrictive in practice and much stricter than those in Brenier's Theorem, including the compactness and convexity of the probability support and the bi-Lipschitz property of the OT maps. We aim to broaden the scope of OT map estimation and fill this gap between theory and practice. Given the strong convexity assumption on Brenier's potential, we first establish the non-asymptotic convergence rates for the original plug-in estimator without requiring restrictive assumptions on probability measures. Additionally, we introduce a sieve plug-in estimator and establish its convergence rates without the strong convexity assumption on Brenier's potential, enabling the widely used cases such as the rank functions of normal or t-distributions. We also establish new Poincar\'e-type inequalities, which are proved given sufficient conditions on the local boundedness of the probability density and mild topological conditions of the support, and these new inequalities enable us to achieve faster convergence rates for the Donsker function class. Moreover, we develop scalable algorithms to efficiently solve the OT map estimation using neural networks and present numerical experiments to demonstrate the effectiveness and robustness.