Sharp comparisons between sliced and standard 1-Wasserstein distances

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.