An Eulerian Perspective on Straight-Line Sampling

Track: Main Track

Keywords: Dynamic measure transport; flow-based models; straight-line flows; displacement interpolation; generative modeling theory.

TL;DR: An Eulerian PDE characterizes when the conditional velocity field of a stochastic process induces a straight (one-step integrable) flow; we fully characterize the affine process case and derive general geometric constraints.

Abstract: 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.

Submission Number: 100