# Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport

arXiv: https://arxiv.org/abs/1905.10812 · HTML: https://mlsanity.com/paper/1905.10812/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 May 2019.

## Built on by (most cited first, 19 of 19)

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- [Optimizing Functionals on the Space of Probabilities with Input Convex Neural Networks](https://mlsanity.com/paper/2106.00774/) (2021)
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- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [Optimal Transport for Measures with Noisy Tree Metric](https://mlsanity.com/paper/2310.13653/) (2023)
- [Cycle Consistent Probability Divergences Across Different Spaces](https://mlsanity.com/paper/2111.11328/) (2021)
- [Order Constraints in Optimal Transport](https://mlsanity.com/paper/2110.07275/) (2021)
- [Permuted and Unlinked Monotone Regression in ℝd: an approach based on mixture modeling and optimal transport](https://mlsanity.com/paper/2201.03528/) (2022)
- [Fast block-coordinate Frank-Wolfe algorithm for semi-relaxed optimal transport](https://mlsanity.com/paper/2103.05857/) (2021)
- [Learning Brenier Potentials with Convex Generative Adversarial Neural Networks](https://mlsanity.com/paper/2504.19779/) (2025)
- [Centered plug-in estimation of Wasserstein distances](https://mlsanity.com/paper/2203.11627/) (2022)
- [Algorithms for Weak Optimal Transport with an Application to Economics](https://mlsanity.com/paper/2205.09825/) (2022)
- [Optimal Transport Barycenter via Nonconvex-Concave Minimax Optimization](https://mlsanity.com/paper/2501.14635/) (2025)
- [From Missing Pieces to Masterpieces: Image Completion with Context-Adaptive Diffusion](https://mlsanity.com/paper/2504.14294/) (2025)
- [OpReg-Boost: Learning to Accelerate Online Algorithms with Operator Regression](https://mlsanity.com/paper/2105.13271/) (2021)
- [Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing](https://mlsanity.com/paper/2510.04579/) (2025)
- [Brenier Isotonic Regression](https://mlsanity.com/paper/2603.10452/) (2026)

## Abstract (arXiv)

Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is comparable to the distance between the original densities. Therefore, optimal transport (OT) can only be used in machine learning if it is substantially regularized. On the other hand, one of the greatest achievements of the OT literature in recent years lies in regularity theory: Caffarelli showed that the OT map between two well behaved measures is Lipschitz, or equivalently when considering 2-Wasserstein distances, that Brenier convex potentials (whose gradient yields an optimal map) are smooth. We propose in this work to draw inspiration from this theory and use regularity as a regularization tool. We give algorithms operating on two discrete measures that can recover nearly optimal transport maps with small distortion, or equivalently, nearly optimal Brenier potentials that are strongly convex and smooth. The problem boils down to solving alternatively a convex QCQP and a discrete OT problem, granting access to the values and gradients of the Brenier potential not only on sampled points, but also out of sample at the cost of solving a simpler QCQP for each evaluation. We propose algorithms to estimate and evaluate transport maps with desired regularity properties, benchmark their statistical performance, apply them to domain adaptation and visualize their action on a color transfer task.