Error Bounds for Flow Matching Methods

TL;DR

The paper derives the first error bounds for fully deterministic flow matching between π_0 and π_1, applicable to data without full support. It links the end-to-end Wasserstein error to the L^2 training error and the Lipschitz constant of the learned velocity, and it bounds the true velocity v^X under a smoothness/regularity assumption, yielding a polynomial dependence on ε when optimally restricting the velocity class. By combining these results, it provides explicit W_2 bounds for deterministic flow matching and offers dimension-aware rates, with specialized bounds for probability flow ODEs that reflect practical sampling schedules. The work thus advances theoretical guarantees for deterministic sampling in flow-based generative models and clarifies the role of data regularity and Gaussian smoothing in stabilizing the sampling process.

Abstract

Score-based generative models are a popular class of generative modelling techniques relying on stochastic differential equations (SDE). From their inception, it was realized that it was also possible to perform generation using ordinary differential equations (ODE) rather than SDE. This led to the introduction of the probability flow ODE approach and denoising diffusion implicit models. Flow matching methods have recently further extended these ODE-based approaches and approximate a flow between two arbitrary probability distributions. Previous work derived bounds on the approximation error of diffusion models under the stochastic sampling regime, given assumptions on the $L^2$ loss. We present error bounds for the flow matching procedure using fully deterministic sampling, assuming an $L^2$ bound on the approximation error and a certain regularity condition on the data distributions.

Theorems & Definitions (39)

• Proposition 1
• Theorem 1
• Proposition 2: Alekseev--Gröbner
• proof : Proof of Theorem \ref{['thm:wassersteinbound']}
• Definition 1
• Lemma 1
• Theorem 2
• proof
• Lemma 2
• proof