# A proof of the Caffarelli contraction theorem via entropic regularization

arXiv: https://arxiv.org/abs/1904.06053 · HTML: https://mlsanity.com/paper/1904.06053/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Apr 2019.

## Built on by (most cited first, 8 of 8)

- [Averaging on the Bures-Wasserstein manifold: dimension-free convergence of gradient descent](https://mlsanity.com/paper/2106.08502/) (2021)
- [Projected Langevin dynamics and a gradient flow for entropic optimal transport](https://mlsanity.com/paper/2309.08598/) (2023)
- [A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm](https://mlsanity.com/paper/2412.09235/) (2024)
- [Tight Lower Bounds for State Tomography with Limited Entanglement](https://mlsanity.com/paper/2609.05718/) (2026)
- [Computational Guarantees for Doubly Entropic Wasserstein Barycenters via Damped Sinkhorn Iterations](https://mlsanity.com/paper/2307.13370/) (2023)
- [Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity](https://mlsanity.com/paper/2504.11133/) (2025)
- [Near-Lipschitz stability of the Kim--Milman flow map](https://mlsanity.com/paper/2606.23383/) (2026)
- [Linear Independent Component Analysis via Optimal Transport](https://mlsanity.com/paper/2607.14081/) (2026)

## Abstract (arXiv)

We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent variational characterization of Lipschitz transport map by the second author and Juillet with a convexity property of optimizers in the dual formulation of the entropy-regularized optimal transport (or Schr"odinger) problem.