One-Shot Generative Flows: Existence and Obstructions
For researchers in generative modeling, this paper provides a theoretical understanding of the limitations of straight-line flows, showing that they exist only for certain distributions.
This paper studies when a generative flow between two distributions can be straight (exactly integrable by first-order methods) under the constraint of independent endpoints. It provides characterizations of straightness, constructs explicit straight flows for Gaussian endpoints, and proves impossibility for targets with well-separated modes.
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.