# Estimation of entropy-regularized optimal transport maps between non-compactly supported measures

arXiv: https://arxiv.org/abs/2311.11934 · HTML: https://mlsanity.com/paper/2311.11934/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 20 Nov 2023.

## Builds on (most cited first, 18 of 18)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Model Fusion via Optimal Transport](https://mlsanity.com/paper/1910.05653/) (2019)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Differentiable Ranks and Sorting using Optimal Transport](https://mlsanity.com/paper/1905.11885/) (2019)
- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [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)
- [On the Efficiency of Entropic Regularized Algorithms for Optimal Transport](https://mlsanity.com/paper/1906.01437/) (2019)
- [A Survey on Optimal Transport for Machine Learning: Theory and Applications](https://mlsanity.com/paper/2106.01963/) (2021)
- [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)
- [Stochastic control liaisons: Richard Sinkhorn meets Gaspard Monge on a Schroedinger bridge](https://mlsanity.com/paper/2005.10963/) (2020)
- [On Rank Energy Statistics via Optimal Transport: Continuity, Convergence, and Change Point Detection](https://mlsanity.com/paper/2302.07964/) (2023)

## Built on by (most cited first, 3 of 3)

- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications](https://mlsanity.com/paper/2410.23602/) (2024)
- [Statistical Rates for Entropic Optimal Transport in the Discrete to SubGaussian Regime](https://mlsanity.com/paper/2609.26647/) (2026)

## Abstract (arXiv)

This paper addresses the problem of estimating entropy-regularized optimal transport (EOT) maps with squared-Euclidean cost between source and target measures that are subGaussian. In the case that the target measure is compactly supported or strongly log-concave, we show that for a recently proposed in-sample estimator, the expected squared L2-error decays at least as fast as O(n-1/3) where n is the sample size. For the general subGaussian case we show that the expected L1-error decays at least as fast as O(n-1/6), and in both cases we have polynomial dependence on the regularization parameter. While these results are suboptimal compared to known results in the case of compactness of both the source and target measures (squared L2-error converging at a rate O(n-1)) and for when the source is subGaussian while the target is compactly supported (squared L2-error converging at a rate O(n-1/2)), their importance lie in eliminating the compact support requirements. The proof technique makes use of a bias-variance decomposition where the variance is controlled using standard concentration of measure results and the bias is handled by T1-transport inequalities along with sample complexity results in estimation of EOT cost under subGaussian assumptions. Our experimental results point to a looseness in controlling the variance terms and we conclude by posing several open problems.