Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem

Mena, Gonzalo; Weed, Jonathan

doi:10.48550/arxiv.1905.11882

Cited by 8 publications

(11 citation statements)

References 21 publications

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“…In particular, building on the recent breakthrough by del Barrio & Loubes (2019) which established a central limit theorem for the Wasserstein distance, we provide a central limit theorem for the barycenter in the 2-Wasserstein space. We also extend this result to a central limit theorem for barycenters with respect to the entropy regularized 2-Wasserstein distance (Cuturi 2013), based on the recently derived central limit theorem for these distances (Mena & Weed 2019). These results are purely statistical and do not relate to inference results on the causal effect.…”

Section: Introductionmentioning

confidence: 76%

“…These approaches introduce an entropic regularization of the optimal transport problem which allows to solve the resulting empirical optimal transport problem efficiently via Sinkhorn iterations. This entropic regularization can be written as (Genevay, Cuturi, Peyré & Bach 2016, Mena & Weed 2019 S ε (P X , P Y ) := inf…”

Section: Methodsmentioning

confidence: 99%

“…The following proposition describes a large sample distribution for this case. It is derived for (11) using and building on the recent result in Mena & Weed (2019). 20 Since uniqueness results for the Sinkhorn barycenter computed by ( 11) have not been established, we derive the large sample distribution for the case of a unique and a non-unique barycenter in the following.…”

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confidence: 99%

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Distributional synthetic controls

Gunsilius¹

2020

Preprint

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This article extends the method of synthetic controls to probability measures. The distribution of the synthetic control group is obtained as the optimally weighted barycenter in Wasserstein space of the distributions of the control groups which minimizes the distance to the distribution of the treatment group. It can be applied to settings with disaggregated-or aggregated (functional) data. The method produces a generically unique counterfactual distribution when the data are continuously distributed. A basic representation of the barycenter provides a computationally efficient implementation via a straightforward tensor-variate regression approach. In addition, identification results are provided that also shed new light on the classical synthetic controls estimator. As an illustration the approach estimates the aggregate effect of the legalization of cannabis on the distribution of household income in Colorado one year after Amendment 64.

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Section: Introductionmentioning

confidence: 76%

Section: Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

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Distributional synthetic controls

Gunsilius¹

2020

Preprint

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“…This line of ideas was put forward in the seminal paper by Cuturi [11], who also emphasizes the smoothing effect, which is crucial when using Sinkhorn as a loss function to train deep learning models. Another benefit of this regularization is that it suffers less from the curse of dimensionality, as proved in [20,41]. This approach is also pivotal to scale our unsupervised metric learning method to tackle for instance applications in genomics.…”

Section: Previous Workmentioning

confidence: 99%

Unsupervised Ground Metric Learning using Wasserstein Singular Vectors

Huizing¹,

Cantini²,

Peyré³

2021

Preprint

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Optimal Transport (OT) defines geometrically meaningful "Wasserstein" distances, used in machine learning applications to compare probability distributions. However, a key bottleneck is the design of a "ground" cost which should be adapted to the task under study. In most cases, supervised metric learning is not accessible, and one usually resorts to some ad-hoc approach. Unsupervised metric learning is thus a fundamental problem to enable data-driven applications of Optimal Transport. In this paper, we propose for the first time a canonical answer by computing the ground cost as a positive eigenvector of the function mapping a cost to the pairwise OT distances between the inputs. This map is homogeneous and monotone, thus framing unsupervised metric learning as a non-linear Perron-Frobenius problem. We provide criteria to ensure the existence and uniqueness of this eigenvector. In addition, we introduce a scalable computational method using entropic regularization, which -in the large regularization limit -operates a principal component analysis dimensionality reduction. We showcase this method on synthetic examples and datasets. Finally, we apply it in the context of biology to the analysis of a high-throughput single-cell RNA sequencing (scRNAseq) dataset, to improve cell clustering and infer the relationships between genes in an unsupervised way.

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“…, in all dimensions (see [25] for sharper results specialized to quadratic cost). Despite this fast convergence in the two-sample test, sample complexity bounds in the (stronger) one-sample regime are not available.…”

Section: Introductionmentioning

confidence: 99%

Gaussian-Smooth Optimal Transport: Metric Structure and Statistical Efficiency

Goldfeld,

Greenewald

2020

Preprint

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Optimal transport (OT), and in particular the Wasserstein distance, has seen a surge of interest and applications in machine learning. However, empirical approximation under Wasserstein distances suffers from a severe curse of dimensionality, rendering them impractical in high dimensions. As a result, entropically regularized OT has become a popular workaround. However, while it enjoys fast algorithms and better statistical properties, it looses the metric structure that Wasserstein distances enjoy. This work proposes a novel Gaussian-smoothed OT (GOT) framework, that achieves the best of both worlds: preserving the 1-Wasserstein metric structure while alleviating the empirical approximation curse of dimensionality. Furthermore, as the Gaussian-smoothing parameter shrinks to zero, GOT Γ-converges towards classic OT (with convergence of optimizers), thus serving as a natural extension. An empirical study that supports the theoretical results is provided, promoting Gaussian-smoothed OT as a powerful alternative to entropic OT.1 Any p-Wasserstein distance has these properties. 2 Note that while Wasserstein-type GANs in practice typically use the two-sample setup since the generator distribution is intractable to compute, fundamentally the GAN actually corresponds to a one-sample setup since infinite samples can be obtained from the generator network.

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Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem

Cited by 8 publications

References 21 publications

Distributional synthetic controls

Distributional synthetic controls

Unsupervised Ground Metric Learning using Wasserstein Singular Vectors

Gaussian-Smooth Optimal Transport: Metric Structure and Statistical Efficiency

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