Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances
This work provides theoretical guarantees for sliced Wasserstein distances, addressing scalability issues in high-dimensional machine learning and statistics, though it is incremental as it builds on existing distance concepts.
The paper tackled the scalability of sliced Wasserstein distances by deriving empirical convergence rates with dimension-dependent constants under log-concavity, showing dimension-free robust estimation risks and equivalence to robust mean estimation, and analyzing computational methods with faster error convergence in higher dimensions and an O(ε^{-4}) complexity bound for max-sliced distance optimization.
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 $O(ε^{-4})$ computational complexity bound for it. Our theory is validated by numerical experiments, which altogether provide a comprehensive quantitative account of the scalability question.