Exploring Diffusion and Flow Matching Under Generator Matching

TL;DR

The paper analyzes diffusion and flow matching through the Generator Matching framework, showing that both are manifestations of Markov processes that connect a simple prior to a target distribution via marginal paths and generators. By recasting these models with the Kolmogorov Forward Equation and generator theory, it derives relationships between forward and reverse dynamics, and formalizes training via a Generator Matching objective that leverages a linear generator parameterization and its linearized CGM variant. A key insight is that diffusion's second-order dynamics introduce instability in backward inference, whereas flow matching's first-order dynamics offer more robust invertibility, a distinction that the Generator Matching view explains and unifies. The work further proposes Markov superposition and potential hybrids with state-dependent diffusion schedules to blend stochastic smoothing and deterministic transport, suggesting practical paths for more robust and data-efficient generative models.

Abstract

In this paper, we present a comprehensive theoretical comparison of diffusion and flow matching under the Generator Matching framework. Despite their apparent differences, both diffusion and flow matching can be viewed under the unified framework of Generator Matching. By recasting both diffusion and flow matching under the same generative Markov framework, we provide theoretical insights into why flow matching models can be more robust empirically and how novel model classes can be constructed by mixing deterministic and stochastic components. Our analysis offers a fresh perspective on the relationships between state-of-the-art generative modeling paradigms.