Title: Flow Matching Generalizes Through Discretization Bias
Primary area: generative models
Keywords: Flow matching, generative model, nonparametric regression, kernel density estimation, implicit bias, ODE discretization

Abstract: Flow models exhibit an extraordinary ability to generalize, generating realistic samples far beyond the training data. This phenomenon lacks a simple explanation. We argue that the key mechanism is not the accurate solution of a continuous-time ODE, but rather the error introduced by its discretization. To isolate this effect within the flow matching framework, we introduce the \emph{Empirical Velocity Field (EVF)}, a non-parametric estimator of the \emph{conditional velocity field} derived by replacing the target distribution with its empirical measure. The exact ODE flow driven by the EVF turns out to be uninteresting, yielding a kernel density estimate that collapses onto the training data. However, its discretization is remarkably powerful. We show that even a single Euler step induces a projection-like effect, concentrating samples on the underlying data manifold and creating diverse, high-quality samples. We support this with extensive empirical evidence and provide a theoretical analysis of the one-step estimator that quantifies this projection, offering a rigorous foundation for how discretization generates structured samples. Our findings argue that the generative success of flow matching is fundamentally driven by the implicit bias of numerical ODE solvers.