mlsanityData to 14 Oct 2026

Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives

arXiv 2506.19025 ↗First posted 23 Jun 2025math.ST cs.AI cs.LGcited by 1

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Abstract (arXiv)

In many applications of optimal transport (OT), the object of primary interest is the optimal transport map. This map rearranges mass from one probability distribution to another in the most efficient way possible by minimizing a specified cost. In this paper we review recent advances in estimating and developing limit theorems for the OT map, using samples from the underlying distributions. We also review parallel lines of work that establish similar results for special cases and variants of the basic OT setup. We conclude with a discussion of key directions for future research with the goal of providing practitioners with reliable inferential tools.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 64 of its 166 references that are in the map. First posted on arXiv 23 Jun 2025.

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64 of 166 references.

References · 64

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  3. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flowcited by 3,700
  4. 2013Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distancescited by 2,316
  5. 2016Input Convex Neural Networkscited by 414
  6. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 292
  7. 2017Near-linear time approximation algorithms for optimal transport via Sinkhorn iterationcited by 223
  8. 2019Optimal transport mapping via input convex neural networkscited by 206
  9. 2018Statistical Aspects of Wasserstein Distancescited by 203
  10. 2019Subspace Robust Wasserstein Distancescited by 90
  11. 2021Entropic estimation of optimal transport mapscited by 90
  12. 2019Statistical bounds for entropic optimal transport: sample complexity and the central limit theoremcited by 85
38 more
  1. 2017Smooth and Sparse Optimal Transportcited by 83
  2. 2021Plugin Estimation of Smooth Optimal Transport Mapscited by 68
  3. 2021Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projectionscited by 53
  4. 2024Statistical optimal transportcited by 47
  5. 2018Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spacescited by 41
  6. 2018Statistical Optimal Transport via Factored Couplingscited by 38
  7. 2021Entropic Optimal Transport: Convergence of Potentialscited by 31
  8. 2018Optimal Transport: Fast Probabilistic Approximation with Exact Solverscited by 27
  9. 2019Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein spacecited by 25
  10. 2016Regularized Optimal Transport and the Rot Mover's Distancecited by 24
  11. 2019Minimax Confidence Intervals for the Sliced Wasserstein Distancecited by 24
  12. 2019Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportationcited by 24
  13. 2016Inference for Empirical Wasserstein Distances on Finite Spacescited by 21
  14. 2022Optimal transport map estimation in general function spacescited by 20
  15. 2023Minimax estimation of discontinuous optimal transport maps: The semi-discrete casecited by 20
  16. 2017Quadratically-Regularized Optimal Transport on Graphscited by 19
  17. 2022An improved central limit theorem and fast convergence rates for entropic transportation costscited by 18
  18. 2019Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptationcited by 17
  19. 2021A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimationcited by 17
  20. 2019A fast approach to optimal transport: The back-and-forth methodcited by 16
  21. 2020Linearized Optimal Transport for Collider Eventscited by 16
  22. 2021Distribution-on-Distribution Regression via Optimal Transport Mapscited by 16
  23. 2016Tsallis Regularized Optimal Transport and Ecological Inferencecited by 15
  24. 2013Minimax rates in permutation estimation for feature matchingcited by 13
  25. 2016Convergence of a Newton algorithm for semi-discrete optimal transportcited by 13
  26. 2018Uncoupled isotonic regression via minimum Wasserstein deconvolutioncited by 12
  27. 2021Near-optimal estimation of smooth transport maps with kernel sums-of-squarescited by 12
  28. 2022Convergence Rates for Regularized Optimal Transport via Quantizationcited by 12
  29. 2023Lower Complexity Adaptation for Empirical Entropic Optimal Transportcited by 12
  30. 2023Minimum intrinsic dimension scaling for entropic optimal transportcited by 11
  31. 2025Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implicationscited by 10
  32. 2022Convergence rate of general entropic optimal transport costscited by 9
  33. 2024Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier mapscited by 9
  34. 2022On the Uniqueness of Kantorovich Potentialscited by 7
  35. 2016Learning relationships between data obtained independentlycited by 6
  36. 2022Background Modeling for Double Higgs Boson Production: Density Ratios and Optimal Transportcited by 6
  37. 2019Disentangled Representation Learning with Wasserstein Total Correlationcited by 5
  38. 2021Matching for causal effects via multimarginal unbalanced optimal transportcited by 4

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  1. 2026The Fundamental Limits of Valid Transport Map Estimationcited by 0