Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections

Deb, Nabarun; Ghosal, Promit; Sen, Bodhisattva

doi:10.48550/arxiv.2107.01718

Cited by 6 publications

(9 citation statements)

References 99 publications

(153 reference statements)

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“…Combining this with ( 41), (42), and the above choice of R then yields the assertion. Note that in the first inequality (43), we have used that B = max 1≤j≤p θ j 2 ≤ max 1≤i≤n y i 2 =: Q since ϕ(z) = ϕ( z 2 ) is decreasing in z 2 .…”

Section: Rates Of Convergence Of the Npmle For Gaussian Location Mixt...mentioning

confidence: 68%

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Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

Slawski¹,

Sen²

2022

Preprint

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Suppose that we have a regression problem with response variable Y ∈ R d and predictor X ∈ R d , for d ≥ 1. In permuted or unlinked regression we have access to separate unordered data on X and Y , as opposed to data on (X, Y )-pairs in usual regression. So far in the literature the case d = 1 has received attention, see e.g., the recent papers by Rigollet and Weed [Information & Inference,8, and Balabdaoui et al. [J. Mach. Learn. Res., 22(172), 1-60]. In this paper, we consider the general multivariate setting with d ≥ 1. We show that the notion of cyclical monotonicity of the regression function is sufficient for identification and estimation in the permuted/unlinked regression model. We study permutation recovery in the permuted regression setting and develop a computationally efficient and easy-to-use algorithm for denoising based on the Kiefer-Wolfowitz [Ann. Math. Statist.,27,[887][888][889][890][891][892][893][894][895][896][897][898][899][900][901][902][903][904][905][906] nonparametric maximum likelihood estimator and techniques from the theory of optimal transport. We provide explicit upper bounds on the associated mean squared denoising error for Gaussian noise. As in previous work on the case d = 1, the permuted/unlinked setting involves slow (logarithmic) rates of convergence rooting in the underlying deconvolution problem. Numerical studies corroborate our theoretical analysis and show that the proposed approach performs at least on par with the methods in the aforementioned prior work in the case d = 1 while achieving substantial reductions in terms of computational complexity.

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Section: Rates Of Convergence Of the Npmle For Gaussian Location Mixt...mentioning

confidence: 68%

“…The approach taken in this paper and the techniques used for its analysis bear various connections to recent developments in the literature on optimal transport, e.g., on the estimation of (smooth) optimal transport maps [40,41,42,43,44]. Key steps in our proofs are based on adaptations of parts of the analysis in [42,43,44].…”

Section: Introductionmentioning

confidence: 99%

Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

Slawski¹,

Sen²

2022

Preprint

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“…During the final stages of preparation of our manuscript, we became aware of the recent independent work of Deb et al (2021) which also studies convergence rates for related plugin estimators of optimal transport maps and Wasserstein distances. A future version of this manuscript will include a more thorough comparison with their work.…”

Section: Concurrent Workmentioning

confidence: 99%

Plugin Estimation of Smooth Optimal Transport Maps

Manole¹,

Balakrishnan²,

Niles-Weed³

et al. 2021

Preprint

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We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived from our observations, appropriately extended so that they define functions on R d . When the underlying map is assumed to be Lipschitz, we show that computing the optimal coupling between the empirical measures, and extending it using linear smoothers, already gives a minimax optimal estimator. When the underlying map enjoys higher regularity, we show that the optimal coupling between appropriate nonparametric density estimates yields faster rates. Our work also provides new bounds on the risk of corresponding plugin estimators for the quadratic Wasserstein distance, and we show how this problem relates to that of estimating optimal transport maps using stability arguments for smooth and strongly convex Brenier potentials. As an application of our results, we derive a central limit theorem for a density plugin estimator of the squared Wasserstein distance, which is centered at its population counterpart when the underlying distributions have sufficiently smooth densities. In contrast to known central limit theorems for empirical estimators, this result easily lends itself to statistical inference for Wasserstein distances.

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“…We note that the soft rank as defined above corresponds to what is referred to as the barycentric projection of optimal transport plan [Seguy et al, 2017, Deb et al, 2021. Based on Definition 3, we now define the population version of soft rank energy.…”

Section: Proposed Soft Rank Energymentioning

confidence: 99%

Multivariate soft rank via entropic optimal transport: sample efficiency and generative modeling

Masud¹,

Werenski²,

Murphy³

et al. 2021

Preprint

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We consider the problem of generating valid knockoffs for knockoff filtering which is a statistical method that provides provable false discovery rate guarantees for any model selection procedure. To this end, we are motivated by recent advances in multivariate distribution-free goodness-of-fit tests namely, the rank energy (RE), that is derived using theoretical results characterizing the optimal maps in the Monge's Optimal Transport (OT) problem. However, direct use of use RE for learning generative models is not feasible because of its high computational and sample complexity, saturation under large support discrepancy between distributions, and non-differentiability in generative parameters. To alleviate these, we begin by proposing a variant of the RE, dubbed as soft rank energy (sRE), and its kernel variant called as soft rank maximum mean discrepancy (sRMMD) using entropic regularization of Monge's OT problem. We then use sRMMD to generate deep knockoffs and show via extensive evaluation that it is a novel and effective method to produce valid knockoffs, achieving comparable, or in some cases improved tradeoffs between detection power Vs false discoveries.

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Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections

Cited by 6 publications

References 99 publications

Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

Permuted and Unlinked Monotone Regression in $\mathbb{R}^d$: an approach based on mixture modeling and optimal transport

Plugin Estimation of Smooth Optimal Transport Maps

Multivariate soft rank via entropic optimal transport: sample efficiency and generative modeling

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