# Near-Lipschitz stability of the Kim--Milman flow map

arXiv: https://arxiv.org/abs/2606.23383 · HTML: https://mlsanity.com/paper/2606.23383/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 46 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space](https://mlsanity.com/paper/1910.05954/) (2019)
- [Minimax estimation of discontinuous optimal transport maps: The semi-discrete case](https://mlsanity.com/paper/2301.11302/) (2023)
- [Projected Langevin dynamics and a gradient flow for entropic optimal transport](https://mlsanity.com/paper/2309.08598/) (2023)
- [Displacement smoothness of entropic optimal transport](https://mlsanity.com/paper/2210.00225/) (2022)
- [Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications](https://mlsanity.com/paper/2502.12326/) (2025)
- [A proof of the Caffarelli contraction theorem via entropic regularization](https://mlsanity.com/paper/1904.06053/) (2019)
- [Stability of optimal transport maps on Riemannian manifolds](https://mlsanity.com/paper/2504.05412/) (2025)
- [Unstable optimal transport maps](https://mlsanity.com/paper/2510.13265/) (2025)
- [Global Regularity Estimates for Optimal Transport via Entropic Regularisation](https://mlsanity.com/paper/2501.11382/) (2025)
- [Stability of the Kim--Milman flow map](https://mlsanity.com/paper/2511.01154/) (2025)
- [Statistical Estimation of Monge Transport Maps via Brenier Potentials](https://mlsanity.com/paper/2604.22366/) (2026)

## Abstract (arXiv)

We prove that the Kim--Milman flow map enjoys favorable stability properties with respect to variations in the target measure, provided that one of the target measures is sufficiently regular. Our results include stability in relative entropy, and more notably, Lipschitz stability in the 2-Wasserstein distance up to a logarithmic factor. We complement our results with a general existence theorem for these maps for any target measure with finite second moment.