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：在紧致设定下，经验最优传输代价向其总体值的收敛速率已在广泛的空间和代价函数类中得到充分理解。然而，在无界设定下，现有结果通常要求对基础代价函数和所涉及测度的集中性施加强假设。本研究提出一种基于分解的方法，将紧致空间中的收敛速率推广到无界设定，所依赖的矩条件在任意小的 $\epsilon > 0$ 范围内是尖锐的。我们证明，紧致空间上经验最优传输的标志性性质（例如最近发现的针对低复杂性的自适应性）可延续至无界情形。