Diffusion Flow Matching: Dimension-Improved KL Bounds and Wasserstein Guarantees

2026-06-15

AI Summary

This work addresses the lack of rigorous theoretical guarantees for existing diffusion flow matching (DFM) methods regarding convergence in Kullback–Leibler (KL) divergence and 2-Wasserstein distance under discretization error. Focusing on Brownian-motion-based DFM, the authors introduce assumptions of finite moments and mild score integrability, combined with a weak log-concavity condition, to establish improved dimension-dependent KL convergence bounds. Notably, this analysis is extended— for the first time—to the 2-Wasserstein distance. Leveraging tools from probability measure theory and functional inequalities, the study achieves state-of-the-art dimension scaling under minimal assumptions, significantly strengthening the theoretical convergence guarantees for DFM.

Abstract

Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood. In this work, we provide refined and novel convergence guarantees for Brownian motion based DFMs, focusing on the discretization error. Our analysis is conducted under the Kullback-Leibler (KL) divergence and the 2-Wasserstein distance. Under finite-moment conditions and a mild score integrability assumption, we derive KL convergence bounds with improved dimensional dependence compared to prior work, achieving, up to our knowledge, state-of-the-art scaling under minimal conditions. We further extend the analysis to the 2-Wasserstein distance: under an additional first-order score integrability assumption and a weak log-concavity condition, we obtain convergence guarantees with dimensional dependence consistent with the KL case.

Problem

• Diffusion Flow Matching
• convergence guarantees
• discretization error
• KL divergence
• Wasserstein distance

Innovation

• Diffusion Flow Matching
• KL divergence
• Wasserstein distance
• dimensional dependence
• convergence guarantees