Research
One-Shot Generative Flows: Existence and Obstructions
arXiv:2604.15439v1 Announce Type: cross Abstract: We study dynamic measure transport for generative modelling in the setting of a stochastic process X_ullet whose marginals interpolate between a sourc
arXiv:2604.15439v1 Announce Type: cross Abstract: We study dynamic measure transport for generative modelling in the setting of a stochastic process X_ullet 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_0otimes P_1. Conditional expectations of this process X_ullet define an ODE whose flow map transports from P_0 to P_1. We discuss when such a process induces a 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.
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Source: arXiv cs.LG | 2026-04-20