We establish weak limits for the empirical entropy regularized optimal transport cost, the expectation of the empirical plan and the conditional expectation. Our results require only uniform boundedness of the cost function and no smoothness properties, thus emphasizing the far-reaching regularizing nature of entropy penalization. To derive these results, we employ a novel technique that sidesteps the intricacies linked to empirical process theory and the control of suprema of function classes determined by the cost. Instead, we perform a careful linearization analysis for entropic optimal transport with respect to an empirical $L^2$-norm, which enables a streamlined analysis. As a consequence, our work gives rise to new implications for a multitude of transport-based applications under general costs, including pointwise distributional limits for the empirical entropic optimal transport map estimator, kernel methods as well as regularized colocalization curves. Overall, our research lays the foundation for an expanded framework of statistical inference with empirical entropic optimal transport.

翻译：我们建立了经验熵正则化最优传输代价、经验计划期望及条件期望的弱极限。研究结果仅需成本函数的一致有界性，无需光滑性假设，从而凸显了熵惩罚深远的正则化本质。为推导这些结果，我们采用了一项新技术，避开了经验过程理论及由成本决定的函数类上确界控制中的复杂问题，转而通过对经验$L^2$范数下的熵正则化最优传输进行精细线性化分析，实现了流程的简化。这一工作为基于传输的广泛应用（包括经验熵正则化最优传输映射估计量的逐点分布极限、核方法以及正则化共定位曲线）在一般成本下带来了新的理论启示。总体而言，我们的研究为基于经验熵正则化最优传输的统计推断框架拓展奠定了基础。