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

arXiv 2606.23383 ↗First posted 22 Jun 2026math.PR math.FA stat.MLcited by 0

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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.

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References · 15

  1. 2013A survey of the Schr"odinger problem and some of its connections with optimal transportcited by 292
  2. 2021Entropic estimation of optimal transport mapscited by 90
  3. 2021Plugin Estimation of Smooth Optimal Transport Mapscited by 68
  4. 2025Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distancecited by 29
  5. 2019Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein spacecited by 25
  6. 2023Minimax estimation of discontinuous optimal transport maps: The semi-discrete casecited by 20
  7. 2023Projected Langevin dynamics and a gradient flow for entropic optimal transportcited by 17
  8. 2022Displacement smoothness of entropic optimal transportcited by 13
  9. 2025Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implicationscited by 10
  10. 2019A proof of the Caffarelli contraction theorem via entropic regularizationcited by 8
  11. 2025Stability of optimal transport maps on Riemannian manifoldscited by 4
  12. 2025Unstable optimal transport mapscited by 2
3 more
  1. 2025Global Regularity Estimates for Optimal Transport via Entropic Regularisationcited by 1
  2. 2025Stability of the Kim--Milman flow mapcited by 1
  3. 2026Statistical Estimation of Monge Transport Maps via Brenier Potentialscited by 1

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