# An improved central limit theorem and fast convergence rates for entropic transportation costs

arXiv: https://arxiv.org/abs/2204.09105 · HTML: https://mlsanity.com/paper/2204.09105/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 80 references that are in the map.
- First posted on arXiv 19 Apr 2022.

## Builds on (most cited first, 30 of 31)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (2016)
- [Wasserstein Distributionally Robust Optimization: Theory and Applications in Machine Learning](https://mlsanity.com/paper/1908.08729/) (2019)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Theoretical Analysis of Domain Adaptation with Optimal Transport](https://mlsanity.com/paper/1610.04420/) (2016)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Faster Wasserstein Distance Estimation with the Sinkhorn Divergence](https://mlsanity.com/paper/2006.08172/) (2020)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [FlipTest: Fairness Testing via Optimal Transport](https://mlsanity.com/paper/1906.09218/) (2019)
- [Entropy-Regularized 2-Wasserstein Distance between Gaussian Measures](https://mlsanity.com/paper/2006.03416/) (2020)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm](https://mlsanity.com/paper/1905.13194/) (2019)
- [Transport-based Counterfactual Models](https://mlsanity.com/paper/2108.13025/) (2021)
- [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021)
- [Wasserstein Discriminant Analysis](https://mlsanity.com/paper/1608.08063/) (2016)
- [Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation](https://mlsanity.com/paper/1909.08733/) (2019)
- [Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2110.06798/) (2021)
- [Inference for Empirical Wasserstein Distances on Finite Spaces](https://mlsanity.com/paper/1610.03287/) (2016)
- [Tackling Algorithmic Bias in Neural-Network Classifiers using Wasserstein-2 Regularization](https://mlsanity.com/paper/1908.05783/) (2019)
- [The statistical effect of entropic regularization in optimal transportation](https://mlsanity.com/paper/2006.05199/) (2020)
- [Asymptotics for semi-discrete entropic optimal transport](https://mlsanity.com/paper/2106.11862/) (2021)
- [Stability of Schr"odinger Potentials and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2201.10059/) (2022)
- [Uncoupled isotonic regression via minimum Wasserstein deconvolution](https://mlsanity.com/paper/1806.10648/) (2018)
- [Multivariate goodness-of-Fit tests based on Wasserstein distance](https://mlsanity.com/paper/2003.06684/) (2020)
- [Fairness with Continuous Optimal Transport](https://mlsanity.com/paper/2101.02084/) (2021)

## Built on by (most cited first, 18 of 18)

- [Trajectory Inference via Mean-field Langevin in Path Space](https://mlsanity.com/paper/2205.07146/) (2022)
- [Energy-guided Entropic Neural Optimal Transport](https://mlsanity.com/paper/2304.06094/) (2023)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [Lower Complexity Adaptation for Empirical Entropic Optimal Transport](https://mlsanity.com/paper/2306.13580/) (2023)
- [Minimum intrinsic dimension scaling for entropic optimal transport](https://mlsanity.com/paper/2306.03398/) (2023)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach](https://mlsanity.com/paper/2304.06549/) (2023)
- [Gaussian Processes on Distributions based on Regularized Optimal Transport](https://mlsanity.com/paper/2210.06574/) (2022)
- [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023)
- [Distributional Counterfactual Explanations With Optimal Transport](https://mlsanity.com/paper/2401.13112/) (2024)
- [Diffeomorphic Measure Matching with Kernels for Generative Modeling](https://mlsanity.com/paper/2402.08077/) (2024)
- [Multivariate Stochastic Dominance via Optimal Transport and Applications to Models Benchmarking](https://mlsanity.com/paper/2406.06425/) (2024)
- [Meta Optimality for Demographic Parity Constrained Regression via Post-Processing](https://mlsanity.com/paper/2506.13947/) (2025)
- [Improved learning theory for kernel distribution regression with two-stage sampling](https://mlsanity.com/paper/2308.14335/) (2023)
- [Leveraging Optimal Transport for Distributed Two-Sample Testing: An Integrated Transportation Distance-based Framework](https://mlsanity.com/paper/2506.16047/) (2025)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transport](https://mlsanity.com/paper/2609.22240/) (2026)

## Abstract (arXiv)

We prove a central limit theorem for the entropic transportation cost between subgaussian probability measures, centered at the population cost. This is the first result which allows for asymptotically valid inference for entropic optimal transport between measures which are not necessarily discrete. In the compactly supported case, we complement these results with new, faster, convergence rates for the expected entropic transportation cost between empirical measures. Our proof is based on strengthening convergence results for dual solutions to the entropic optimal transport problem.