Stability of the Kim--Milman flow map

TL;DR

This work studies the stability of the Kim–Milman (probability flow) transport map from the standard Gaussian $\gamma$ to a target measure $\mu$, with respect to perturbations of the target measured by the relative Fisher information $FI(\nu\|\mu)$. The authors derive an explicit FI-based contraction bound for the reverse heat flow and translate it into an $L^2(\gamma)$ stability bound $\|T_{\rm KM}^{\mu} - T_{\rm KM}^{\nu}\|_{L^2(\gamma)} \lesssim \sqrt{FI(\nu\|\mu)}$, plus an $L^{\infty}$ bound under stronger conditions involving $FI_{\infty}(\nu\|\mu)$. They instantiate the result in several concrete settings, including when $\mu$ is $\alpha$-strongly log-concave, when $\mu$ is a log-Lipschitz perturbation of a strongly log-concave measure, and for distributions with asymptotically positive convex profiles, providing explicit constants. An extension to stronger metrics is developed, yielding $L^{\infty}$-type stability and a transport-information inequality. Overall, the paper establishes the first FI-based stability guarantees for the Kim–Milman flow map and highlights regimes not covered by stability results for (entropic) OT maps.

Abstract

In this short note, we characterize stability of the Kim--Milman flow map -- also known as the probability flow ODE -- with respect to variations in the target measure. Rather than the Wasserstein distance, we show that stability holds with respect to the relative Fisher information

Theorems & Definitions (12)

• Proposition 3.1
• Theorem 3.2
• Lemma 3.3: wibisono2025mixing
• Corollary 3.4
• Corollary 3.5: Perturbation of strongly log-concave
• Proposition 3.6: conforti2024weak
• Corollary 3.7: Asymptotically positive convex profile
• Lemma 4.1
• Proposition 4.2
• Theorem 4.3