We obtain essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from $n$ samples. By proving a Banach space version of this result, we also obtain an upper bound, that is sharp up to a log factor, for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure $\mu$ on a Euclidean space and its symmetrized empirical distribution in terms of the norm of the covariance matrix of $\mu$ and the diameter of the support of $\mu$.

翻译：我们得到了可分希尔伯特空间上概率测度与其基于$n$个样本的经验分布之间的期望最大切片1-Wasserstein距离的本质上匹配的上下界。通过证明该结果的巴拿赫空间版本，我们还得到了欧几里得空间上对称概率测度$\mu$与其对称化经验分布之间的期望最大切片2-Wasserstein距离的上界，该上界在最多一个对数因子意义下是尖锐的，且以$\mu$的协方差矩阵范数和$\mu$支撑集的直径为表述。