# Estimation of Stochastic Optimal Transport Maps

arXiv: https://arxiv.org/abs/2512.09499 · HTML: https://mlsanity.com/paper/2512.09499/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 50 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Builds on (most cited first, 25 of 25)

- [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)
- [Style Transfer by Relaxed Optimal Transport and Self-Similarity](https://mlsanity.com/paper/1904.12785/) (2019)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [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)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [Generalized Resilience and Robust Statistics](https://mlsanity.com/paper/1909.08755/) (2019)
- [Monge-Amp`ere Flow for Generative Modeling](https://mlsanity.com/paper/1809.10188/) (2018)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [Kernel Neural Optimal Transport](https://mlsanity.com/paper/2205.15269/) (2022)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Wasserstein Style Transfer](https://mlsanity.com/paper/1905.12828/) (2019)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [Large-scale optimal transport map estimation using projection pursuit](https://mlsanity.com/paper/2106.05838/) (2021)
- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications](https://mlsanity.com/paper/2502.12326/) (2025)
- [Robust W-GAN-Based Estimation Under Wasserstein Contamination](https://mlsanity.com/paper/2101.07969/) (2021)
- [Robust Estimation under the Wasserstein Distance](https://mlsanity.com/paper/2302.01237/) (2023)
- [Improved Rate of First Order Algorithms for Entropic Optimal Transport](https://mlsanity.com/paper/2301.09675/) (2023)
- [GAN Estimation of Lipschitz Optimal Transport Maps](https://mlsanity.com/paper/2202.07965/) (2022)
- [Optimal Robust Estimation under Local and Global Corruptions: Stronger Adversary and Smaller Error](https://mlsanity.com/paper/2410.17230/) (2024)
- [Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures](https://mlsanity.com/paper/2503.10576/) (2025)

## Built on by (most cited first, 1 of 1)

- [UniMergeTrack: Hierarchical Model Merging for General RGB-X Visual Object Tracking](https://mlsanity.com/paper/2610.13053/) (2026)

## Abstract (arXiv)

The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. However, existing statistical theory for OT map estimation is quite restricted, hinging on Brenier's theorem (quadratic cost, absolutely continuous source) to guarantee existence and uniqueness of a deterministic OT map, on which various additional regularity assumptions are imposed to obtain quantitative error bounds. In many real-world problems these conditions fail or cannot be certified, in which case optimal transportation is possible only via stochastic maps that can split mass. To broaden the scope of map estimation theory to such settings, this work introduces a novel metric for evaluating the transportation quality of stochastic maps. Under this metric, we develop computationally efficient map estimators with near-optimal finite-sample risk bounds, subject to easy-to-verify minimal assumptions. Our analysis further accommodates common forms of adversarial sample contamination, yielding estimators with robust estimation guarantees. Empirical experiments are provided which validate our theory and demonstrate the utility of the proposed framework in settings where existing theory fails. These contributions constitute the first general-purpose theory for map estimation, compatible with a wide spectrum of real-world applications where optimal transport may be intrinsically stochastic.