# Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps

arXiv: https://arxiv.org/abs/2411.07154 · HTML: https://mlsanity.com/paper/2411.07154/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 50 references that are in the map.
- First posted on arXiv 11 Nov 2024.

## Builds on (most cited first, 30 of 31)

- [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)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Conditional Image Generation with Score-Based Diffusion Models](https://mlsanity.com/paper/2111.13606/) (2021)
- [Kernel Operations on the GPU, with Autodiff, without Memory Overflows](https://mlsanity.com/paper/2004.11127/) (2020)
- [Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein](https://mlsanity.com/paper/2201.12324/) (2022)
- [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023)
- [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)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Conditional Sampling with Monotone GANs: from Generative Models to Likelihood-Free Inference](https://mlsanity.com/paper/2006.06755/) (2020)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [Conditional Optimal Transport on Function Spaces](https://mlsanity.com/paper/2311.05672/) (2023)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Causal transport in discrete time and applications](https://mlsanity.com/paper/1606.04062/) (2016)
- [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)
- [Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference](https://mlsanity.com/paper/2310.16975/) (2023)
- [On minimax density estimation via measure transport](https://mlsanity.com/paper/2207.10231/) (2022)
- [Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation](https://mlsanity.com/paper/1905.10155/) (2019)
- [Triangular Flows for Generative Modeling: Statistical Consistency, Smoothness Classes, and Fast Rates](https://mlsanity.com/paper/2112.15595/) (2021)
- [Neural Optimal Transport with Lagrangian Costs](https://mlsanity.com/paper/2406.00288/) (2024)
- [Computational methods for adapted optimal transport](https://mlsanity.com/paper/2203.05005/) (2022)
- [Minimum intrinsic dimension scaling for entropic optimal transport](https://mlsanity.com/paper/2306.03398/) (2023)
- [A generative flow for conditional sampling via optimal transport](https://mlsanity.com/paper/2307.04102/) (2023)

## Built on by (most cited first, 9 of 9)

- [A friendly introduction to triangular transport](https://mlsanity.com/paper/2503.21673/) (2025)
- [Modeling Stochastic Conditional Dynamics from Sparse Observations via Kernel-Stabilized Flow Matching](https://mlsanity.com/paper/2411.08314/) (2024)
- [A Geometric Approach to Optimal Experimental Design](https://mlsanity.com/paper/2510.14848/) (2025)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Forward Reverse Kernel Regression for the Schr"odinger bridge problem](https://mlsanity.com/paper/2507.00640/) (2025)
- [Trajectory inference via Acceleration Matching](https://mlsanity.com/paper/2608.03916/) (2026)
- [Energy-based Transport for Amortized Bayesian Inference](https://mlsanity.com/paper/2605.15407/) (2026)
- [Nesterov acceleration in optimizing over probability measures](https://mlsanity.com/paper/2607.23008/) (2026)
- [Approximating Measures on Function Spaces: Transport and Truncation](https://mlsanity.com/paper/2609.17802/) (2026)

## Abstract (arXiv)

Conditional simulation is a fundamental task in statistical modeling: Generate samples from the conditionals given finitely many data points from a joint distribution. One promising approach is to construct conditional Brenier maps, where the components of the map pushforward a reference distribution to conditionals of the target. While many estimators exist, few, if any, come with statistical or algorithmic guarantees. To this end, we propose a non-parametric estimator for conditional Brenier maps based on the computational scalability of entropic optimal transport. Our estimator leverages a result of Carlier et al. (2010), which shows that optimal transport maps under a rescaled quadratic cost asymptotically converge to conditional Brenier maps; our estimator is precisely the entropic analogues of these converging maps. We provide heuristic justifications for choosing the scaling parameter in the cost as a function of the number of samples by fully characterizing the Gaussian setting. We conclude by comparing the performance of the estimator to other machine learning and non-parametric approaches on benchmark datasets and Bayesian inference problems.