# Entropic estimation of optimal transport maps

arXiv: https://arxiv.org/abs/2109.12004 · HTML: https://mlsanity.com/paper/2109.12004/

## Key facts

- Cited by 90 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 66 references that are in the map.
- First posted on arXiv 24 Sep 2021.

## Builds on (most cited first, 22 of 22)

- [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)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [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)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [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)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Massively scalable Sinkhorn distances via the Nystr"om method](https://mlsanity.com/paper/1812.05189/) (2018)
- [Optimal Transport for Diffeomorphic Registration](https://mlsanity.com/paper/1706.05218/) (2017)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021)
- [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021)
- [Learning normalizing flows from Entropy-Kantorovich potentials](https://mlsanity.com/paper/2006.06033/) (2020)
- [Asymptotics for semi-discrete entropic optimal transport](https://mlsanity.com/paper/2106.11862/) (2021)

## Built on by (most cited first, 30 of 90)

- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Debias Coarsely, Sample Conditionally: Statistical Downscaling through Optimal Transport and Probabilistic Diffusion Models](https://mlsanity.com/paper/2305.15618/) (2023)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics](https://mlsanity.com/paper/2310.09254/) (2023)
- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [Sinkhorn-Drifting Generative Models](https://mlsanity.com/paper/2603.12366/) (2026)
- [Multivariate Conformal Prediction using Optimal Transport](https://mlsanity.com/paper/2502.03609/) (2025)
- [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)
- [An improved central limit theorem and fast convergence rates for entropic transportation costs](https://mlsanity.com/paper/2204.09105/) (2022)
- [Light Schr"odinger Bridge](https://mlsanity.com/paper/2310.01174/) (2023)
- [Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps](https://mlsanity.com/paper/2302.04065/) (2023)
- [Neural Optimal Transport with Lagrangian Costs](https://mlsanity.com/paper/2406.00288/) (2024)
- [Rethinking Initialization of the Sinkhorn Algorithm](https://mlsanity.com/paper/2206.07630/) (2022)
- [Near-optimal estimation of smooth transport maps with kernel sums-of-squares](https://mlsanity.com/paper/2112.01907/) (2021)
- [Convergence Rates for Regularized Optimal Transport via Quantization](https://mlsanity.com/paper/2208.14391/) (2022)
- [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)
- [Mirror and Preconditioned Gradient Descent in Wasserstein Space](https://mlsanity.com/paper/2406.08938/) (2024)
- [Minimum intrinsic dimension scaling for entropic optimal transport](https://mlsanity.com/paper/2306.03398/) (2023)
- [FlashSinkhorn: IO-Aware Entropic Optimal Transport on GPU](https://mlsanity.com/paper/2602.03067/) (2026)
- [Progressive Entropic Optimal Transport Solvers](https://mlsanity.com/paper/2406.05061/) (2024)
- [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)
- [Light Unbalanced Optimal Transport](https://mlsanity.com/paper/2303.07988/) (2023)
- [Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2311.04041/) (2023)
- [Structured Transforms Across Spaces with Cost-Regularized Optimal Transport](https://mlsanity.com/paper/2311.05788/) (2023)

## Abstract (arXiv)

We develop a computationally tractable method for estimating the optimal map between two distributions over ℝd with rigorous finite-sample guarantees. Leveraging an entropic version of Brenier's theorem, we show that our estimator -- the barycentric projection of the optimal entropic plan -- is easy to compute using Sinkhorn's algorithm. As a result, unlike current approaches for map estimation, which are slow to evaluate when the dimension or number of samples is large, our approach is parallelizable and extremely efficient even for massive data sets. Under smoothness assumptions on the optimal map, we show that our estimator enjoys comparable statistical performance to other estimators in the literature, but with much lower computational cost. We showcase the efficacy of our proposed estimator through numerical examples, even ones not explicitly covered by our assumptions. By virtue of Lepski's method, we propose a modified version of our estimator that is adaptive to the smoothness of the underlying optimal transport map. Our proofs are based on a modified duality principle for entropic optimal transport and on a method for approximating optimal entropic plans due to Pal (2019).