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Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances
Sloan Nietert · Ziv Goldfeld · Ritwik Sadhu · Kengo Kato

Tue Nov 29 02:00 PM -- 04:00 PM (PST) @ Hall J #435
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.

Author Information

Sloan Nietert (Cornell University)
Ziv Goldfeld (Cornell University)
Ritwik Sadhu (Cornell University)
Ritwik Sadhu

Ritwik Sadhu is a Ph.D. Candidate at the Department of Statistics and Data Science, Cornell University. His research interests lie at the intersection of nonparametric inference and optimal transport, including central limit theorems and convergence rates for optimal transport distances, applications of these distances to nonparametric testing problems, as well as high dimensional signal detection threshold problems.

Kengo Kato (Cornell University)

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