# Optimal transport map estimation in general function spaces arXiv: https://arxiv.org/abs/2212.03722 ยท HTML: https://mlsanity.com/paper/2212.03722/ ## Key facts - Cited by 20 papers in the map (data through 14 Oct 2026). - Builds on 18 of its 55 references that are in the map. - First posted on arXiv 7 Dec 2022. ## Builds on (most cited first, 18 of 18) - [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022) - [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018) - [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014) - [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017) - [A Review on Generative Adversarial Networks: Algorithms, Theory, and Applications](https://mlsanity.com/paper/2001.06937/) (2020) - [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019) - [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018) - [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020) - [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021) - [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021) - [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021) - [Supervised Training of Conditional Monge Maps](https://mlsanity.com/paper/2206.14262/) (2022) - [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021) - [Optimal Transport for Diffeomorphic Registration](https://mlsanity.com/paper/1706.05218/) (2017) - [Learning normalizing flows from Entropy-Kantorovich potentials](https://mlsanity.com/paper/2006.06033/) (2020) - [Concentration bounds for linear Monge mapping estimation and optimal transport domain adaptation](https://mlsanity.com/paper/1905.10155/) (2019) - [Near-optimal estimation of smooth transport maps with kernel sums-of-squares](https://mlsanity.com/paper/2112.01907/) (2021) - [An Optimal Transport Approach to Estimating Causal Effects via Nonlinear Difference-in-Differences](https://mlsanity.com/paper/2108.05858/) (2021) ## Built on by (most cited first, 20 of 20) - [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) - [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023) - [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) - [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023) - [Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps](https://mlsanity.com/paper/2411.07154/) (2024) - [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024) - [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024) - [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023) - [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) - [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) - [Meta Optimality for Demographic Parity Constrained Regression via Post-Processing](https://mlsanity.com/paper/2506.13947/) (2025) - [Error Analysis of Bayesian Inverse Problems with Generative Priors](https://mlsanity.com/paper/2601.17374/) (2026) - [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025) - [The Fundamental Limits of Valid Transport Map Estimation](https://mlsanity.com/paper/2606.30574/) (2026) - [Deep Barycentric Regression for Optimal Transport Map Estimation and its Statistical Optimality](https://mlsanity.com/paper/2609.06598/) (2026) - [Improving Function Space Flow Matching with Kernel Optimal Transport](https://mlsanity.com/paper/2609.38049/) (2026) ## Abstract (arXiv) We study the problem of estimating a function T given independent samples from a distribution P and from the pushforward distribution T_sharp P. This setting is motivated by applications in the sciences, where T represents the evolution of a physical system over time, and in machine learning, where, for example, T may represent a transformation learned by a deep neural network trained for a generative modeling task. To ensure identifiability, we assume that T = nabla varphi0 is the gradient of a convex function, in which case T is known as an optimal transport map. Prior work has studied the estimation of T under the assumption that it lies in a H"older class, but general theory is lacking. We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure P satisfy a Poincar'e inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for H"older transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when P is the normal distribution and the transport map is given by an infinite-width shallow neural network.