One-Shot Generative Flows: Existence and Obstructions

Authors	Panos Tsimpos et al.
Year	2026
Field	Statistics / ML
arXiv	2604.15439
PDF	Download
Categories	stat.ML, cs.LG

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.

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Engineering Breakdown

Plain English

This paper studies how to transport probability distributions using stochastic processes where the source and target distributions remain independent throughout the flow. The authors characterize when such transport processes induce straight-line flows—paths whose acceleration vanishes and can be exactly solved by simple first-order numerical methods. They develop PDE-based characterizations of this straightness property and prove a sharp dichotomy exists: under endpoint independence constraints, straightness either holds cleanly or faces fundamental obstructions. This result matters because straight-line flows are computationally cheaper to implement and integrate than curved paths, potentially reducing sampling cost in generative modeling.

Core Technical Contribution

The key novelty is formalizing and characterizing straight-line flows in the setting of independent endpoint marginals (where X₀ ~ P₀ and X₁ ~ P₁ are independent). Unlike prior work on generative flows that assumes coupling between source and target, this paper proves that straightness under independence exhibits a dichotomy—a sharp bifurcation where straightness either exists provably or faces fundamental mathematical obstructions. The authors provide multiple PDE-based characterizations linking straightness to conditional statistics of the stochastic process, moving beyond heuristic flow designs to exact structural conditions. This is the first rigorous treatment of when one-shot transport (independent endpoints) can yield exact, integrable dynamics.

How It Works

The paper starts with a stochastic process X_• whose marginals smoothly interpolate from source P₀ to target P₁ while maintaining independence: (X₀, X₁) ~ P₀ ⊗ P₁. The conditional expectation field E[X_t | X₀, X₁] defines an ODE that governs the flow map transporting mass from P₀ to P₁. Straightness is defined by the pointwise acceleration vanishing—meaning the velocity field of the flow is constant along trajectories, making it exactly solvable via Euler's method. The authors derive PDE constraints (involving conditional covariances and cross-moments) that characterize when this straightness property holds. They then prove that under the independence constraint, these PDEs force a dichotomy: either the process satisfies a rigid structural condition (e.g., Gaussian marginals with specific scaling), or it provably cannot be straight.

Production Impact

In production generative modeling pipelines, straight-line flows reduce the cost of sampling and probability estimation because they require no ODE solver—a single forward pass with constant velocity suffices. If you adopt this approach, you gain exact numerical integration (no truncation error from adaptive solvers), deterministic trajectories (no randomness in the flow itself), and lower memory overhead during inference. The trade-off is restrictive: you must ensure your data admits the structural conditions for straightness (e.g., your source and target distributions must be truly independent, and their conditional statistics must satisfy the PDE constraints). This is most practical for controlled settings like diffusion model baselines where you can design P₀ and P₁ carefully, rather than arbitrary data distributions. For standard use cases where you want flexibility in choosing source and target, the obstructions theorem may force you back to curved flows.

Limitations and When Not to Use This

The main limitation is the dichotomy itself: straightness under independence is highly constrained, and many practical distributions will violate the necessary PDE conditions, leaving you with no straight path. The paper assumes endpoint independence, which is stronger than typical generative flow setups where source and target can be coupled—relaxing this may dissolve the sharp dichotomy. The characterizations are theoretical and don't yet provide algorithmic guidance on how to verify or construct processes satisfying the straightness conditions for arbitrary data distributions. Open questions remain about whether relaxing independence slightly, or allowing discrete-time approximations, can recover some benefits of straightness in broader settings.

Research Context

This work builds on the recent wave of optimal transport and flow-based generative models (Flow Matching, Rectified Flows) that aim to design transport paths with minimal computational cost. It directly engages with the question posed by Albergo & Vanden-Eijnden on whether straight-line flows are always possible—answering that they are not, except under restrictive conditions. The paper advances the theoretical foundations of measure transport by proving obstructions, similar to how topological results constrain what flows are achievable. This opens a new research direction: characterizing achievable flow geometries under different independence or coupling assumptions, and developing algorithms that gracefully degrade to curved flows when straightness is obstructed.

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