  Sliced Wasserstein distances preserve properties of classic Wasserstein
distances while being more scalable for computation and estimation in high
dimensions. The goal of this work is to quantify this scalability from three
key aspects: (i) empirical convergence rates; (ii) robustness to data
contamination; and (iii) efficient computational methods. For empirical
convergence, we derive fast rates with explicit dependence of constants on
dimension, subject to log-concavity of the population distributions. For
robustness, we characterize minimax optimal, dimension-free robust estimation
risks, and show an equivalence between robust sliced 1-Wasserstein estimation
and robust mean estimation. This enables lifting statistical and algorithmic
guarantees available for the latter to the sliced 1-Wasserstein setting. Moving
on to computational aspects, we analyze the Monte Carlo estimator for the
average-sliced distance, demonstrating that larger dimension can result in
faster convergence of the numerical integration error. For the max-sliced
distance, we focus on a subgradient-based local optimization algorithm that is
frequently used in practice, albeit without formal guarantees, and establish an
$O(\epsilon^{-4})$ computational complexity bound for it. Our theory is
validated by numerical experiments, which altogether provide a comprehensive
quantitative account of the scalability question.
