# Near-optimal estimation of smooth transport maps with kernel sums-of-squares

arXiv: https://arxiv.org/abs/2112.01907 · HTML: https://mlsanity.com/paper/2112.01907/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 37 references that are in the map.
- First posted on arXiv 3 Dec 2021.

## Builds on (most cited first, 17 of 17)

- [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)
- [Joint Distribution Optimal Transportation for Domain Adaptation](https://mlsanity.com/paper/1705.08848/) (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)
- [Improving GANs Using Optimal Transport](https://mlsanity.com/paper/1803.05573/) (2018)
- [Less is More: Nystr"om Computational Regularization](https://mlsanity.com/paper/1507.04717/) (2015)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [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)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [Optimal Transport for Diffeomorphic Registration](https://mlsanity.com/paper/1706.05218/) (2017)
- [Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport](https://mlsanity.com/paper/1905.10812/) (2019)
- [A Dimension-free Computational Upper-bound for Smooth Optimal Transport Estimation](https://mlsanity.com/paper/2101.05380/) (2021)
- [On parameter estimation with the Wasserstein distance](https://mlsanity.com/paper/1701.05146/) (2017)

## Built on by (most cited first, 12 of 12)

- [Wasserstein Flow Matching: Generative modeling over families of distributions](https://mlsanity.com/paper/2411.00698/) (2024)
- [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)
- [Measure Estimation in the Barycentric Coding Model](https://mlsanity.com/paper/2201.12195/) (2022)
- [Learning Elastic Costs to Shape Monge Displacements](https://mlsanity.com/paper/2306.11895/) (2023)
- [Semi-supervised Learning of Pushforwards For Domain Translation & Adaptation](https://mlsanity.com/paper/2304.08673/) (2023)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport](https://mlsanity.com/paper/2310.14087/) (2023)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Stochastic optimal transport in Banach Spaces for regularized estimation of multivariate quantiles](https://mlsanity.com/paper/2302.00982/) (2023)
- [GloptiNets: Scalable Non-Convex Optimization with Certificates](https://mlsanity.com/paper/2306.14932/) (2023)
- [Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis](https://mlsanity.com/paper/2605.24644/) (2026)

## Abstract (arXiv)

It was recently shown that under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. However, rather than the distance itself, the object of interest for applications such as generative modeling is the underlying optimal transport map. Hence, computational and statistical guarantees need to be obtained for the estimated maps themselves. In this paper, we propose the first tractable algorithm for which the statistical L2 error on the maps nearly matches the existing minimax lower-bounds for smooth map estimation. Our method is based on solving the semi-dual formulation of optimal transport with an infinite-dimensional sum-of-squares reformulation, and leads to an algorithm which has dimension-free polynomial rates in the number of samples, with potentially exponentially dimension-dependent constants.