Minimax confidence intervals for the Sliced Wasserstein distance

Manole, Tudor; Balakrishnan, Sivaraman; Wasserman, Larry

doi:10.1214/22-ejs2001

Cited by 15 publications

(17 citation statements)

References 57 publications

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“…Recent work have bounded E|SW p (µ n , ν n ; ρ) − SW p (µ, ν; ρ)| or E|SW p (µ, µ n ; ρ)| for specific choices of ρ ∈ P(S d−1 ) [18,46,33,41]. These results do not exactly correspond to what Theorem 2 requires, i.e.…”

Section: A3 Proof Of Propositionmentioning

confidence: 98%

“…We move on to our second setting for distributions with unbounded supports, which relaxes A2 by assuming a Bernstein-type moment condition instead, as described hereafter. Proposition 2 [46] Table 1: Overview of the explicit forms of ϕ µ,ν,p and ψ µ,ν,p under different assumptions.…”

Section: Distributions Supported On Unbounded Domainsmentioning

confidence: 99%

“…Indeed, since the supports of µ, ν are unbounded, the necessary condition for the finiteness of {SJ(µ; ρ) + SJ(ν; ρ)} is not met, thus deteriorating the rate of convergence in Proposition 2. To overcome this issue, we will use the rate recently established in [46], which holds true on A2 and A3 and shows that ψ µ,ν,p scales as n −1/2 log(n). Our final bound is obtained by plugging in Theorem 2 the explicit formula of ψ µ,ν,1 and Propositions 3 and 4: we present this result and its detailed proof in Appendix A.7.…”

Section: Distributions Supported On Unbounded Domainsmentioning

confidence: 99%

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Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances

Ohana¹,

Nadjahi²,

Rakotomamonjy³

et al. 2022

Preprint

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The Sliced-Wasserstein distance (SW) is a computationally efficient and theoretically grounded alternative to the Wasserstein distance. Yet, the literature on its statistical properties with respect to the distribution of slices, beyond the uniform measure, is scarce. To bring new contributions to this line of research, we leverage the PAC-Bayesian theory and the central observation that SW actually hinges on a slice-distribution-dependent Gibbs risk, the kind of quantity PAC-Bayesian bounds have been designed to characterize. We provide four types of results: i) PAC-Bayesian generalization bounds that hold on what we refer as adaptive Sliced-Wasserstein distances, i.e. distances defined with respect to any distribution of slices, ii) a procedure to learn the distribution of slices that yields a maximally discriminative SW, by optimizing our PAC-Bayesian bounds, iii) an insight on how the performance of the so-called distributional Sliced-Wasserstein distance may be explained through our theory, and iv) empirical illustrations of our findings.

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Section: A3 Proof Of Propositionmentioning

confidence: 98%

Section: Distributions Supported On Unbounded Domainsmentioning

confidence: 99%

Section: Distributions Supported On Unbounded Domainsmentioning

confidence: 99%

See 1 more Smart Citation

Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances

Ohana¹,

Nadjahi²,

Rakotomamonjy³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Notably, the centering term is the expected empirical distance as opposed to the population one. This gap was addressed in [61], where the asymptotic normality of √ n W 2 2 ( P n , Q) − W 2 2 (P, Q) was established for distributions P and Q with sufficiently smooth densities. Their estimator P n is the probability measure associated with a wavelet-based density estimate constructed from samples from P .…”

Section: Related Workmentioning

confidence: 99%

Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications

Ritwik

Goldfeld

Kato

2021

Preprint

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The smooth 1-Wasserstein distance (SWD) W σ 1 was recently proposed as a means to mitigate the curse of dimensionality in empirical approximation, while preserving the Wasserstein structure. Indeed, SWD exhibits parametric convergence rates and inherits the metric and topological structure of the classic Wasserstein distance. Motivated by the above, this work conducts a thorough statistical study of the SWD, including a high-dimensional limit distribution result for empirical W σ 1 , bootstrap consistency, concentration inequalities, and Berry-Esseen type bounds. The derived nondegenerate limit stands in sharp contrast with the classic empirical W 1 , for which a similar result is known only in the one-dimensional case. We also explore asymptotics and characterize the limit distribution when the smoothing parameter σ is scaled with n, converging to 0 at a sufficiently slow rate. The dimensionality of the sampled distribution enters empirical SWD convergence bounds only through the prefactor (i.e., the constant). We provide a sharp characterization of this prefactor's dependence on the smoothing parameter and the intrinsic dimension. This result is then used to derive new empirical convergence rates for classic W 1 in terms of the intrinsic dimension. As applications of the limit distribution theory, we study two-sample testing and minimum distance estimation (MDE) under W σ 1 . We establish asymptotic validity of SWD testing, while for MDE, we prove measurability, almost sure convergence, and limit distributions for optimal estimators and their corresponding W σ 1 error. Our results suggest that the SWD is well suited for high-dimensional statistical learning and inference.

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“…In this line of research, [15] showed that, for Gaussian measures, the sample complexity of SW has at most polynomial dependency on the dimension of the problem d, compared to the sample complexity of the Wasserstein distance, which has an exponential dependence on d [19]. [20] then refined that result, and showed that the convergence rate of empirical measures under SW does not depend on the dimension, assuming some moment conditions on the measures. Finally, [21] showed that the estimators obtained by minimizing SW converge to a true estimator with a rate of n −1/2 , thus independent of d, where n denotes the number of observed samples.…”

Section: Introductionmentioning

confidence: 99%

Statistical and Topological Properties of Sliced Probability Divergences

Nadjahi¹,

Durmus²,

Chizat³

et al. 2020

Preprint

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The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a 'base divergence' between one-dimensional random projections of the two measures. However, the computational and statistical consequences of such a technique have not yet been well-established. In this paper, we aim at bridging this gap and derive some properties of sliced divergence functions. First, we show that slicing preserves the metric axioms and the weak continuity of the divergence, implying that the sliced divergence will share similar topological properties. We then precise the results in the case where the base divergence belongs to the class of integral probability metrics. On the other hand, we establish that, under mild conditions, the sample complexity of the sliced divergence does not depend on the dimension, even when the base divergence suffers from the curse of dimensionality. We finally apply our general results to the Wasserstein distance and Sinkhorn divergences, and illustrate our theory on both synthetic and real data experiments.

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Minimax confidence intervals for the Sliced Wasserstein distance

Cited by 15 publications

References 57 publications

Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances

Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances

Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications

Statistical and Topological Properties of Sliced Probability Divergences

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