One-Shot Generative Flows: Existence and Obstructions

2026-04-16

AI Summary

This work investigates whether “straight” generative flows—those realizable via a single forward pass—can transform a source distribution into a target distribution under endpoint-independent conditions. By constructing a stochastic process that interpolates between the two distributions, the authors derive an ordinary differential equation (ODE) flow through its conditional expectation and characterize straightness via zero acceleration. The study reveals a sharp dichotomy: explicit straight flows exist for any pair of Gaussian endpoints, yet no such flow can exist for sufficiently separated multimodal targets. The analysis establishes a profound connection between sample-path behavior and the spatiotemporal geometry of the flow map, provides multiple equivalent partial differential equation (PDE) characterizations, and delineates the precise existence boundary of straight generative flows through an impossibility theorem.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Computer Vision: Diffusion Models for Vision

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

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.

Problem

• generative flows
• measure transport
• straight-line flow
• independent endpoints
• multimodal distributions

Innovation

• straight-line flow
• dynamic measure transport
• independent endpoints
• generative modeling
• impossibility theorems