An Eulerian Perspective on Straight-Line Sampling
This work provides theoretical guidance for designing more easily integrable transport processes in generative modeling, representing an incremental advance in the field.
The paper tackles the problem of identifying which stochastic processes produce straight-line flows in dynamic measure transport for generative modeling, showing that straightness occurs exactly under deterministic endpoint couplings and providing necessary conditions that constrain flow geometry for general processes.
We study dynamic measure transport for generative modeling: specifically, flows induced by stochastic processes that bridge a specified source and target distribution. The conditional expectation of the process' velocity defines an ODE whose flow map achieves the desired transport. We ask \emph{which processes produce straight-line flows} -- i.e., flows whose pointwise acceleration vanishes and thus are exactly integrable with a first-order method? We provide a concise PDE characterization of straightness as a balance between conditional acceleration and the divergence of a weighted covariance (Reynolds) tensor. Using this lens, we fully characterize affine-in-time interpolants and show that straightness occurs exactly under deterministic endpoint couplings. We also derive necessary conditions that constrain flow geometry for general processes, offering broad guidance for designing transports that are easier to integrate.