This paper introduces the first statistically consistent estimator of the optimal transport map between two probability distributions, based on neural networks. Building on theoretical and practical advances in the field of Lipschitz neural networks, we define a Lipschitz-constrained generative adversarial network penalized by the quadratic transportation cost. Then, we demonstrate that, under regularity assumptions, the obtained generator converges uniformly to the optimal transport map as the sample size increases to infinity. Furthermore, we show through a number of numerical experiments that the learnt mapping has promising performances. In contrast to previous work tackling either statistical guarantees or practicality, we provide an expressive and feasible estimator which paves way for optimal transport applications where the asymptotic behaviour must be certified.

翻译：本文件介绍了第一个统计上一致的根据神经网络在两种概率分布之间最佳运输图的最佳估计数据。我们根据Lipschitz神经网络领域的理论和实践进步,定义了受二次运输成本约束的Lipschitz受限制的基因对抗网络。然后,我们证明,根据常规假设,获得的生成器与最佳运输图一致,因为样本大小增加至无限。此外,我们通过一些数字实验表明,所学的绘图有良好的性能。与以往处理统计保障或实用性的工作相比,我们提供了明确可行的估计数据,为最佳运输应用铺平了道路,必须验证无症状行为。