Measure estimation on manifolds: an optimal transport approach

Divol, Vincent

doi:10.1007/s00440-022-01118-z

Cited by 7 publications

(1 citation statement)

References 46 publications

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“…For example, if this measure possesses a lower and upper bounded density on some bounded subset of R d , then the rate is known to be n −q/d even if q > d/2 [21]. This rate may even be improved if μ X concentrates on a low-dimensional submanifold of R d [39,16], which is particularly relevant in the UQ context which motivates this study, see Remark 5.3. In order to make the use of our results as flexible as possible, from now on we shall denote by (τ q,d (n)) n≥1 a sequence such that E W q q (μ X , μ Xn ) = O (τ q,d (n)) , and thus…”

Section: Convergence Of μ (K)mentioning

confidence: 92%

Reweighting samples under covariate shift using a Wasserstein distance criterion

Reygner

Touboul

2022

Electron. J. Statist.

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Considering two random variables with different laws to which we only have access through finite size i.i.d samples, we address how to reweight the first sample so that its empirical distribution converges towards the true law of the second sample as the size of both samples goes to infinity. We study an optimal reweighting that minimizes the Wasserstein distance between the empirical measures of the two samples, and leads to an expression of the weights in terms of Nearest Neighbors. The consistency and some asymptotic convergence rates in terms of expected Wasserstein distance are derived, and do not need the assumption of absolute continuity of one random variable with respect to the other. These results have some application in Uncertainty Quantification for decoupled estimation and in the bound of the generalization error for the Nearest Neighbor regression under covariate shift.

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Section: Convergence Of μ (K)mentioning

confidence: 92%