Theoretical guarantees in KL for Diffusion Flow Matching

TL;DR

This paper provides theoretical guarantees in KL divergence for Diffusion Flow Matching models, establishing bounds under mild assumptions on distributions and score functions, advancing understanding of their convergence properties.

Contribution

It introduces non-asymptotic KL divergence bounds for DFM models with Brownian motion bridges under mild distributional assumptions.

Findings

Bounds on KL divergence between target and generated distributions.

Conditions under which DFM models reliably approximate target distributions.

Theoretical insights into the convergence behavior of diffusion-based generative models.

Abstract

Flow Matching (FM) (also referred to as stochastic interpolants or rectified flows) stands out as a class of generative models that aims to bridge in finite time the target distribution $\nu^\star$ with an auxiliary distribution $\mu$, leveraging a fixed coupling $\pi$ and a bridge which can either be deterministic or stochastic. These two ingredients define a path measure which can then be approximated by learning the drift of its Markovian projection. The main contribution of this paper is to provide relatively mild assumptions on $\nu^\star$, $\mu$ and $\pi$ to obtain non-asymptotics guarantees for Diffusion Flow Matching (DFM) models using as bridge the conditional distribution associated with the Brownian motion. More precisely, we establish bounds on the Kullback-Leibler divergence between the target distribution and the one generated by such DFM models under moment conditions on the score of $\nu^\star$, $\mu$ and $\pi$, and a standard $L^2$-drift-approximation error assumption.