Minimax Confidence Intervals for the Sliced Wasserstein Distance

Manole, Tudor; Balakrishnan, Sivaraman; Wasserman, Larry

doi:10.48550/arxiv.1909.07862

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…Faster rates of convergence for estimating optimal transport costs are achievable under strong conditions on X and Y. For instance, the bound ∆ n (c) n −1/2 is known to hold when c is a metric raised to a power p ≥ 1, and when X and Y are one-dimensional (Munk and Czado, 1998;Freitag and Munk, 2005;del Barrio et al, 2019b;Manole et al, 2019) or countable (Sommerfeld and Munk, 2018;Tameling et al, 2019). In both of these cases, the corresponding empirical p-Wasserstein distance is known to exhibit distinct convergence rates depending on whether µ and ν are vanishingly close or not, similarly as our findings in equation ( 5).…”

Section: Related Workmentioning

confidence: 99%

Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs

Manole¹,

Niles-Weed²

2021

Preprint

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We revisit the question of characterizing the convergence rate of plug-in estimators of optimal transport costs. It is well known that an empirical measure comprising independent samples from an absolutely continuous distribution on R d converges to that distribution at the rate n −1/d in Wasserstein distance, which can be used to prove that plug-in estimators of many optimal transport costs converge at this same rate. However, we show that when the cost is smooth, this analysis is loose: plug-in estimators based on empirical measures converge quadratically faster, at the rate n −2/d . As a corollary, we show that the Wasserstein distance between two distributions is significantly easier to estimate when the measures are far apart. We also prove lower bounds, showing not only that our analysis of the plug-in estimator is tight, but also that no other estimator can enjoy significantly faster rates of convergence uniformly over all pairs of measures. Our proofs rely on empirical process theory arguments based on tight control of L 2 covering numbers for locally Lipschitz and semi-concave functions. As a byproduct of our proofs, we derive L ∞ estimates on the displacement induced by the optimal coupling between any two measures satisfying suitable moment conditions, for a wide range of cost functions.

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Section: Related Workmentioning

confidence: 99%

Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs

Manole¹,

Niles-Weed²

2021

Preprint

Self Cite

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“…In front of the difficulty to estimate W 2 2 (µ, ν), researchers have also turned their attention to similar but more tractable discrepancy measures such as the sliced Wasserstein distance [47] or the Sinkhorn divergence [48], which can be both estimated at the parametric rate [25,38,37,40]. However, there is "no free lunch" and unconditional statistical efficiency comes at the price of lack of adaptivity and discriminative power.…”

Section: Introductionmentioning

confidence: 99%

Faster Wasserstein Distance Estimation with the Sinkhorn Divergence

Chizat,

Roussillon,

Léger

et al. 2020

Preprint

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The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem. It can be solved to ε-accuracy by adding an entropic regularization of order ε and using for instance Sinkhorn's algorithm. In this work, we propose instead to estimate it with the Sinkhorn divergence, which is also built on entropic regularization but includes debiasing terms. We show that, for smooth densities, this estimator has a comparable sample complexity but allows higher regularization levels, of order ε 1/2 , which leads to improved computational complexity bounds and a strong speedup in practice. Our theoretical analysis covers the case of both randomly sampled densities and deterministic discretizations on uniform grids. We also propose and analyze an estimator based on Richardson extrapolation of the Sinkhorn divergence which enjoys improved statistical and computational efficiency guarantees, under a condition on the regularity of the approximation error, which is in particular satisfied for Gaussian densities. We finally demonstrate the efficiency of the proposed estimators with numerical experiments.

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Minimax Confidence Intervals for the Sliced Wasserstein Distance

Cited by 2 publications

References 26 publications

Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs

Sharp Convergence Rates for Empirical Optimal Transport with Smooth Costs

Faster Wasserstein Distance Estimation with the Sinkhorn Divergence

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