Sliced Wasserstein distances are widely used in practice as a computationally efficient alternative to Wasserstein distances in high dimensions. In this paper, motivated by theoretical foundations of this alternative, we prove quantitative estimates between the sliced $1$-Wasserstein distance and the $1$-Wasserstein distance. We construct a concrete example to demonstrate the exponents in the estimate is sharp. We also provide a general analysis for the case where slicing involves projections onto $k$-planes and not just lines.

翻译：切片 Wasserstein 距离因其在高维情形下的计算高效性，在实践中被广泛用作 Wasserstein 距离的替代。本文基于这一替代方案的理论基础，证明了切片 $1$-Wasserstein 距离与 $1$-Wasserstein 距离之间的定量估计。我们构造了一个具体示例，以证明该估计中的指数是最优的。此外，我们还对切片涉及 $k$ 维平面（而不仅仅是直线）投影的情形进行了通用分析。