Stability of the Kim--Milman flow map

2025-11-02

AI Summary

This paper investigates the stability of Kim–Milman flow maps—i.e., probability flow ODEs—under perturbations of the target measure. Departing from conventional Wasserstein-distance-based frameworks, it introduces relative Fisher information as a novel stability metric, enabling a finer sensitivity analysis grounded in information geometry. Methodologically, the work integrates tools from information geometry, probability flow ODE theory, and measure convergence analysis to derive quantitative stability estimates for the flow map with respect to variations in the target measure. The main contribution is a rigorous proof of continuous dependence of the Kim–Milman flow map on the target measure in the relative Fisher information topology. This establishes the first stability theorem for probability flow maps formulated in terms of information divergence, thereby providing a new paradigm for robustness analysis of probabilistic flow models.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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

Problem

• Characterizing stability of Kim-Milman flow map
• Analyzing variations in target measure effects
• Establishing stability via relative Fisher information

Innovation

• Characterizing stability of Kim-Milman flow map
• Using relative Fisher information metric
• Analyzing variations in target measure