In compact settings, the convergence rate of the empirical optimal transport cost to its population value is well understood for a wide class of spaces and cost functions. In unbounded settings, however, hitherto available results require strong assumptions on the ground costs and the concentration of the involved measures. In this work, we pursue a decomposition-based approach to generalize the convergence rates found in compact spaces to unbounded settings under generic moment assumptions that are sharp up to an arbitrarily small $\epsilon > 0$. Hallmark properties of empirical optimal transport on compact spaces, like the recently established adaptation to lower complexity, are shown to carry over to the unbounded case.

翻译：在紧致设置中，经验最优输运代价向其总体值的收敛速率对于一大类空间和代价函数已有充分理解。然而，在无界设置下，现有结果要求对基础代价和所涉及测度的集中性施加强假设。本文采用一种基于分解的方法，将在紧致空间中建立的收敛速率推广到无界设置，且仅需在泛型矩假设下达到任意小$\epsilon > 0$内精确的收敛速率。本文证明紧致空间上经验最优输运的标志性性质（如近期发现的低复杂度适应性）可推广至无界情形。