A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models

TL;DR

This paper unifies various continuous-time generative modeling methods—diffusion, score-based, and flow matching—under a measure-theoretic framework, clarifying their shared structure, differences, and practical tradeoffs.

Contribution

It provides a unified theoretical framework connecting diffusion, score-based, and flow matching models, deriving new insights into their sampling, training, and relationships.

Findings

Reverse-time sampling derived as controlled stochastic dynamics.

Probability flow ODE connects diffusion models to normalizing flows.

Flow matching interpreted as direct velocity field regression.

Abstract

We survey continuous-time generative modeling methods based on transporting a simple reference distribution to a data distribution via stochastic or deterministic dynamics. We present a unified framework in which diffusion models, score-based generative models, and flow matching are instances of learning a time-dependent vector field that induces a family of marginals $(\rho_t)_{t \in [0,1]}$ governed by continuity and Fokker-Planck equations. Such a unified theory is timely because these methods are converging methodologically, yet fragmented notation and competing derivations continue to obscure their shared structure and the practical tradeoffs governing sampling, stability, and computation. Within this framework, we (i) derive reverse-time sampling for diffusion and score-based models as controlled stochastic dynamics, (ii) show that the probability flow ODE yields identical marginals and connects diffusion to likelihood-based normalizing flows, and (iii) interpret flow matching as direct regression of the velocity field under a chosen interpolation, clarifying when it coincides with or differs from score-based training. We compare objectives, sampling schemes, and discretization errors under unified notation, discuss connections to Schrodinger bridges and entropic optimal transport, and summarize theoretical guarantees and open problems on approximation, stability, and scalability.