Abstract
We study dynamic measure transport for generative modelling in the setting of a stochastic process X_\bullet whose marginals interpolate between a source distribution P_0 and a target distribution P_1 while remaining independent, i.e., when (X_0,X_1)\sim P_0\otimes P_1.
Conditional expectations of this process X_\bullet define an ODE whose flow map transports from P_0 to P_1. We discuss when such a process induces a \emph{straight-line flow}, namely one whose pointwise acceleration vanishes and is therefore exactly integrable by any first-order method.
We first develop multiple characterizations of straightness in terms of PDEs involving the conditional statistics of the process.
Then, we prove that straightness under endpoint independence exhibits a sharp dichotomy.
On one hand, we construct explicit, computable straight-line processes for arbitrary Gaussian endpoints. On the other hand, we show straight-line processes do not exist for targets with sufficiently well-separated modes. We demonstrate this 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 generative flows can, and cannot, exist.
