One-Shot Generative Flows: Existence and Obstructions

We study dynamic measure transport for generative modeling, focusing on transport maps that connect a source measure P0 to a target measure P1 by integrating a velocity field of the form vt(x) = mathbbE[dot Xt mid Xt = x], where X_bullet = (Xt)t is a stochastic process satisfying (X0,X1)~P0otimesP1 and dot Xt is its time derivative. We investigate when X_bullet induces a straight-line flow: a flow whose pointwise acceleration vanishes and is therefore exactly integrable by any first-order method. First, we develop multiple characterizations of straight-line flows in terms of PDEs involving the conditional statistics of the process. Then, we prove that straight-line flows under endpoint independence exhibit a sharp dichotomy. On the one hand, we construct explicit, computable straight-line processes for arbitrary Gaussian endpoints. On the other hand, we show that straight-line processes do not exist for targets with sufficiently well-separated modes. We demonstrate this obstruction through a sequence of increasingly general impossibility theorems that uncover a fundamental relationship between the sample-path behavior of a process with independent endpoints and the space-time geometry of this process' flow map. Taken together, these results provide a structural theory of when straight-line generative flows can, and cannot, exist.