Some remarks on the connection between generative diffusion and optimal transport
Authors: Giulio Biroli, Marc Mézard
Organizations: Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France · Department of Computing Sciences, Bocconi University, Milano, Italy
We discuss three simple connections between generative diffusion and optimal transport. First, the joint law of the data and of their Ornstein-Uhlenbeck noising is, at every time, the solution of an entropic optimal-transport problem, at a temperature set by the noise level. Backward diffusion can consequently be read as an annealing which gradually decreases the entropic regularization. Second, we study the speciation transition--how the modes of the data distribution emerge during generation--and compare how it happens in diffusion and flow matching with respect to optimal transport. Using Gaussian mixtures as high-dimensional example, we show that the dynamical instability through which diffusion and flow matching separate the modes becomes, for optimal transport, an exponentially thin boundary layer at the beginning of generation, which is rounded by entropic regularization. Third, considering a generic forward dynamics with a general confining potential, we show that the backward process driven by the score also satisfies a drift-based variational principle whose minimal cost is the entropy difference between initial and final distributions.
Figures & tables
Figure 1: Backward generative trajectories of the active coordinate for the mixture ( 26 ) ( m=2.2 , σ=0.6 ), integrated from noise to data. Left, optimal transport: straight displacement-interpolation lines; the fan is already open at t=1 , separation taking place within the boundary layer τOT=Θ(Λ−1) of ( 32 ). Center, flow matching (independent coupling, linear clock): curved “tuning fork”, with the bifurcation at tlin≃m/(1+m) (dashed line), cf. ( 28 ). Right, diffusion (independent coupling, OU clock): the same bifurcation plotted against the rescaled forward OU time t/tf , which runs from 1 at the noise end ( tf=4 is the horizon of the simulation) to 0 at the data end; the dashed line marks the speciation time tOU≃21log(1+m2) , i.e. t/tf≃0.22 . The logarithmic clock dilates the boundary layer into a macroscopic window, cf. ( 29 ). The three panels show the same pair of measures transported with different coupling and clock choices.
These notes recapitulate the high level mathematical principles behind different techniques for generative modeling. I show the connections between optimal transport and standard techniques such as Schr{ö}dinger bridge and flow matching.
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 (ρt)t∈[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.
Aditya Ranganath, Mukesh Singhal
Center for Applied Scientific Computing Lawrence Livermore National Laboratory Livermore, CA 94551, USA · Department of Electrical Engineering University of California Merced, CA 95343, USA
Flow matching models learn to transport samples from a simple prior distribution to a complex data distribution. When prior-data pairs are coupled via optimal transport (OT), the learned trajectories are straight and non-crossing, enabling fast, even single-step, generation. However, computing the OT coupling in high dimensions is intractable, and existing methods attempt to solve the OT problem, at the cost of persistent bias or significant overhead. Rather than solving for the OT coupling, we reformulate the problem. Once the prior is treated as a design choice rather than a fixed input, the OT coupling between prior and data is no longer unique. Many priors admit an OT-optimal identity coupling to the data, leaving us free to choose one that is also tractable to sample. We identify low-frequency projection of natural images as such a choice. The identity coupling between data and its low-frequency representation is empirically OT-optimal, the prior is structured enough to be sampled by a lightweight model at inference, and the remaining flow-matching task reduces to synthesizing high-frequency detail. Interpolating the prior with Gaussian noise further improves generation quality while preserving the OT coupling. The approach requires no modifications to the flow model itself, and integrates naturally with latent-space models, classifier-free guidance, and one-step generation frameworks. Across all benchmarks, our method reduces trajectory curvature by more than 2× compared to existing flow matching methods, yielding better generation quality in the few-step regime.