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

arXiv: https://arxiv.org/abs/2107.01718 · HTML: https://mlsanity.com/paper/2107.01718/

## Key facts

- Cited by 53 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 161 references that are in the map.
- First posted on arXiv 4 Jul 2021.

## Builds on (most cited first, 27 of 27)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (2017)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [DeepJDOT: Deep Joint Distribution Optimal Transport for Unsupervised Domain Adaptation](https://mlsanity.com/paper/1803.10081/) (2018)
- [Learning with a Wasserstein Loss](https://mlsanity.com/paper/1506.05439/) (2015)
- [Neural Architecture Search with Bayesian Optimisation and Optimal Transport](https://mlsanity.com/paper/1802.07191/) (2018)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Leveraging Labeled and Unlabeled Data for Consistent Fair Binary Classification](https://mlsanity.com/paper/1906.05082/) (2019)
- [Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance](https://mlsanity.com/paper/1805.11897/) (2018)
- [Regularized Discrete Optimal Transport](https://mlsanity.com/paper/1307.5551/) (2013)
- [Scalable Bayes via Barycenter in Wasserstein Space](https://mlsanity.com/paper/1508.05880/) (2015)
- [Statistical Optimal Transport via Factored Couplings](https://mlsanity.com/paper/1806.07348/) (2018)
- [Stochastic Wasserstein Barycenters](https://mlsanity.com/paper/1802.05757/) (2018)
- [Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation](https://mlsanity.com/paper/1909.08733/) (2019)
- [Nonparametric independence testing via mutual information](https://mlsanity.com/paper/1711.06642/) (2017)
- [Inference for Empirical Wasserstein Distances on Finite Spaces](https://mlsanity.com/paper/1610.03287/) (2016)
- [Procrustes Metrics on Covariance Operators and Optimal Transportation of Gaussian Processes](https://mlsanity.com/paper/1801.01990/) (2018)
- [Fr'echet Means and Procrustes Analysis in Wasserstein Space](https://mlsanity.com/paper/1701.06876/) (2017)
- [Optimal transport: discretization and algorithms](https://mlsanity.com/paper/2003.00855/) (2020)
- [On parameter estimation with the Wasserstein distance](https://mlsanity.com/paper/1701.05146/) (2017)

## Built on by (most cited first, 30 of 53)

- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Optimal transport map estimation in general function spaces](https://mlsanity.com/paper/2212.03722/) (2022)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Neural Optimal Transport with Lagrangian Costs](https://mlsanity.com/paper/2406.00288/) (2024)
- [Near-optimal estimation of smooth transport maps with kernel sums-of-squares](https://mlsanity.com/paper/2112.01907/) (2021)
- [Linearized Wasserstein dimensionality reduction with approximation guarantees](https://mlsanity.com/paper/2302.07373/) (2023)
- [Lower Complexity Adaptation for Empirical Entropic Optimal Transport](https://mlsanity.com/paper/2306.13580/) (2023)
- [Measure Estimation in the Barycentric Coding Model](https://mlsanity.com/paper/2201.12195/) (2022)
- [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024)
- [Learning of Population Dynamics: Inverse Optimization Meets JKO Scheme](https://mlsanity.com/paper/2506.01502/) (2025)
- [Light Unbalanced Optimal Transport](https://mlsanity.com/paper/2303.07988/) (2023)
- [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023)
- [Manifold learning in Wasserstein space](https://mlsanity.com/paper/2311.08549/) (2023)
- [Concept Heterogeneity-aware Representation Steering](https://mlsanity.com/paper/2603.02237/) (2026)
- [Categorical Schr"odinger Bridge Matching](https://mlsanity.com/paper/2502.01416/) (2025)
- [Permuted and Unlinked Monotone Regression in ℝd: an approach based on mixture modeling and optimal transport](https://mlsanity.com/paper/2201.03528/) (2022)
- [Tangential Wasserstein Projections](https://mlsanity.com/paper/2207.14727/) (2022)
- [Diffeomorphic Measure Matching with Kernels for Generative Modeling](https://mlsanity.com/paper/2402.08077/) (2024)
- [Optimal Aggregation of Prediction Intervals under Unsupervised Domain Shift](https://mlsanity.com/paper/2405.10302/) (2024)
- [Trade-off Between Dependence and Complexity for Nonparametric Learning -- an Empirical Process Approach](https://mlsanity.com/paper/2401.08978/) (2024)
- [Optimal Transport for Domain Adaptation through Gaussian Mixture Models](https://mlsanity.com/paper/2403.13847/) (2024)
- [Centered plug-in estimation of Wasserstein distances](https://mlsanity.com/paper/2203.11627/) (2022)
- [A New Perspective On Denoising Based On Optimal Transport](https://mlsanity.com/paper/2312.08135/) (2023)
- [A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers](https://mlsanity.com/paper/2502.01310/) (2025)

## Abstract (arXiv)

Optimal transport maps between two probability distributions μ and nu on ℝd have found extensive applications in both machine learning and statistics. In practice, these maps need to be estimated from data sampled according to μ and nu. Plug-in estimators are perhaps most popular in estimating transport maps in the field of computational optimal transport. In this paper, we provide a comprehensive analysis of the rates of convergences for general plug-in estimators defined via barycentric projections. Our main contribution is a new stability estimate for barycentric projections which proceeds under minimal smoothness assumptions and can be used to analyze general plug-in estimators. We illustrate the usefulness of this stability estimate by first providing rates of convergence for the natural discrete-discrete and semi-discrete estimators of optimal transport maps. We then use the same stability estimate to show that, under additional smoothness assumptions of Besov type or Sobolev type, wavelet based or kernel smoothed plug-in estimators respectively speed up the rates of convergence and significantly mitigate the curse of dimensionality suffered by the natural discrete-discrete/semi-discrete estimators. As a by-product of our analysis, we also obtain faster rates of convergence for plug-in estimators of W2(μ,nu), the Wasserstein distance between μ and nu, under the aforementioned smoothness assumptions, thereby complementing recent results in Chizat et al. (2020). Finally, we illustrate the applicability of our results in obtaining rates of convergence for Wasserstein barycenters between two probability distributions and obtaining asymptotic detection thresholds for some recent optimal-transport based tests of independence.