Entropic optimal transport: convergence of potentials

Nutz, Marcel; Wiesel, Johannes

doi:10.1007/s00440-021-01096-8

Cited by 32 publications

(30 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From a mathematical perspective, this problem and its regularity properties have been analyzed thoroughly, see for instance Carlier & Laborde (2020), Carlier (2021), Genevay et al (2019), di Marino & Gerolin (2020 for recent results and references therein. Note that as ε → 0 the optimal matching γ * solving (2) approximates the optimal matching obtained via the classical Kantorovich problem (1) under mild regularity assumptions (Nutz & Wiesel 2021).…”

Section: Entropy Regularized Optimal Transport and The Schrödinger Br...mentioning

confidence: 91%

Matching for causal effects via multimarginal optimal transport

Gunsilius¹,

Xu²

2021

Preprint

View full text Add to dashboard Cite

Matching on covariates is a well-established framework for estimating causal effects in observational studies. The principal challenge in these settings stems from the often high-dimensional structure of the problem. Many methods have been introduced to deal with this challenge, with different advantages and drawbacks in computational and statistical performance and interpretability. Moreover, the methodological focus has been on matching two samples in binary treatment scenarios, but a dedicated method that can optimally balance samples across multiple treatments has so far been unavailable. This article introduces a natural optimal matching method based on entropy-regularized multimarginal optimal transport that possesses many useful properties to address these challenges. It provides interpretable weights of matched individuals that converge at the parametric rate to the optimal weights in the population, can be efficiently implemented via the classical iterative proportional fitting procedure, and can even match several treatment arms simultaneously. It also possesses demonstrably excellent finite sample properties.

show abstract

Section: Entropy Regularized Optimal Transport and The Schrödinger Br...mentioning

confidence: 91%

Matching for causal effects via multimarginal optimal transport

Gunsilius¹,

Xu²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…As a byproduct of our analysis, by sending ε ց 0 and appealing to recent convergence results for the entropic Brenier potentials [NW21], we obtain the shortest proof of Caffarelli's contraction theorem to date. Notably, our argument allows us to sidestep the regularity of the optimal transport map, which is a key obstacle in Caffarelli's original proof.…”

Section: Introductionmentioning

confidence: 87%

“…The constraint that π ε has marginals P and Q implies the following dual optimality conditions for (f ε , g ε ) (see [MNW19,NW21]):…”

Section: Optimal Transport With Entropic Regularizationmentioning

confidence: 99%

See 1 more Smart Citation

An entropic generalization of Caffarelli's contraction theorem via covariance inequalities

Chewi¹,

Pooladian²

2022

Preprint

View full text Add to dashboard Cite

The optimal transport map between the standard Gaussian measure and an α-strongly logconcave probability measure is α −1/2 -Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp-Lieb and Cramér-Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli's original result. We also extend Caffarelli's theorem to the setting in which the Hessians of the log-densities of the measures are bounded by arbitrary positive definite commuting matrices.

show abstract

“…Most notably, Cuturi [22] proposed to use entropic regularization and the Sinkhorn algorithm for solving the optimal transport problem (i.e., when N = 2). See also [27,47] for the theoretical properties of entropic regularization and the Sinkhorn algorithm. While most regularizationbased approaches deal with discrete marginals, see, e.g., [9,49,56], there are also regularization-based approaches for solving MMOT problems with non-discrete marginals.…”

Section: Introductionmentioning

confidence: 99%

Numerical method for feasible and approximately optimal solutions of multi-marginal optimal transport beyond discrete measures

Neufeld¹,

Xiang²

2022

Preprint

View full text Add to dashboard Cite

We propose a numerical algorithm for the computation of multi-marginal optimal transport (MMOT) problems involving general measures that are not necessarily discrete. By developing a relaxation scheme in which marginal constraints are replaced by finitely many linear constraints and by proving a specifically tailored duality result for this setting, we approximate the MMOT problem by a linear semiinfinite optimization problem. Moreover, we are able to recover a feasible and approximately optimal solution of the MMOT problem, and its sub-optimality can be controlled to be arbitrarily close to 0 under mild conditions. The developed relaxation scheme leads to a numerical algorithm which can compute a feasible approximate optimizer of the MMOT problem whose theoretical sub-optimality can be chosen to be arbitrarily small. Besides the approximate optimizer, the algorithm is also able to compute both an upper bound and a lower bound on the optimal value of the MMOT problem. The difference between the computed bounds provides an explicit upper bound on the sub-optimality of the computed approximate optimizer. Through a numerical example, we demonstrate that the proposed algorithm is capable of computing a high-quality solution of an MMOT problem involving as many as 50 marginals along with an explicit estimate of its sub-optimality that is much less conservative compared to the theoretical estimate.

show abstract

Entropic optimal transport: convergence of potentials

Cited by 32 publications

References 21 publications

Matching for causal effects via multimarginal optimal transport

Matching for causal effects via multimarginal optimal transport

An entropic generalization of Caffarelli's contraction theorem via covariance inequalities

Numerical method for feasible and approximately optimal solutions of multi-marginal optimal transport beyond discrete measures

Contact Info

Product

Resources

About