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

arXiv:2606.23383 · math.PR, math.FA, stat.ML · Submitted 2026-06-22 · Read on arXiv

Sinho Chewi, Katharina Eichinger, Aram-Alexandre Pooladian

math.PR, math.FA, stat.ML

Submitted: 2026-06-22

Comments: 28 pages

License: http://creativecommons.org/licenses/by/4.0/

The gist: 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.

Terminology

• Optimal Transport — 26× in this paper · explained in 5 episodes
• Wasserstein Distance — 13× in this paper · explained in 6 episodes
• Fisher Information — 12× in this paper · explained in 4 episodes
• 2-Wasserstein Distance — 8× in this paper · explained in 1 episode
• Generalization — 5× in this paper · explained in 14 episodes
• Langevin Dynamics — 4× in this paper · explained in 3 episodes
• Regularization — 4× in this paper · explained in 3 episodes
• Editor — 2× in this paper · explained in 2 episodes
• Monotonicity — 2× in this paper · explained in 2 episodes
• Absorb — 1× in this paper · explained in 1 episode
• Composition — 1× in this paper · explained in 2 episodes
• Core — 1× in this paper · explained in 2 episodes
• Diffusion Processes — 1× in this paper · explained in 2 episodes
• Generative Model — 1× in this paper · explained in 2 episodes
• Generative Models — 1× in this paper · explained in 3 episodes
• Gradient Flow — 1× in this paper · explained in 2 episodes
• Lyapunov function — 1× in this paper · explained in 2 episodes
• Sample Complexity — 1× in this paper · explained in 20 episodes

Abstract

Sources

• Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications
• Statistical Estimation of Monge Transport Maps via Brenier Potentials
• Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations
• Stability of the Kim--Milman flow map
• A coupling approach to Lipschitz transport maps
• A Generalization of Caffarelli's Contraction Theorem to Nearly Spherical Manifolds
• Global Regularity Estimates for Optimal Transport via Entropic Regularisation
• Stability of optimal transport maps on Riemannian manifolds
• Gluing methods for quantitative stability of optimal transport maps
• Lipschitz changes of variables via heat flow
• Entropic estimation of optimal transport maps

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