Optimal transport (OT) serves as a natural framework for comparing probability measures, with applications in statistics, machine learning, and applied mathematics. Alas, statistical estimation and exact computation of the OT distances suffer from the curse of dimensionality. To circumvent these issues, entropic regularization has emerged as a remedy that enables parametric estimation rates via plug-in and efficient computation using Sinkhorn iterations. Motivated by further scaling up entropic OT (EOT) to data dimensions and sample sizes that appear in modern machine learning applications, we propose a novel neural estimation approach. Our estimator parametrizes a semi-dual representation of the EOT distance by a neural network, approximates expectations by sample means, and optimizes the resulting empirical objective over parameter space. We establish non-asymptotic error bounds on the EOT neural estimator of the cost and optimal plan. Our bounds characterize the effective error in terms of neural network size and the number of samples, revealing optimal scaling laws that guarantee parametric convergence. The bounds hold for compactly supported distributions and imply that the proposed estimator is minimax-rate optimal over that class. Numerical experiments validating our theory are also provided.

翻译：最优传输（OT）作为比较概率度量的自然框架，在统计学、机器学习和应用数学等领域具有广泛应用。然而，OT距离的统计估计与精确计算受维数灾难的影响。为规避这些问题，熵正则化作为一种解决方案应运而生，它通过插值法实现参数化估计速率，并利用Sinkhorn迭代进行高效计算。为进一步将熵最优传输（EOT）扩展至现代机器学习应用中常见的数据维度与样本规模，我们提出了一种新颖的神经估计方法。该方法通过神经网络参数化EOT距离的半对偶表示，利用样本均值近似期望，并在参数空间上优化所得经验目标函数。我们建立了EOT神经估计器在代价函数与最优传输方案上的非渐近误差界。该误差界以神经网络规模与样本数量表征有效误差，揭示了确保参数化收敛的最优缩放律。误差界适用于紧支集分布，并表明所提估计器在该分布类上达到极小极大速率最优。最后，我们通过数值实验验证了理论结果。