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One-Shot Generative Flows: Existence and Obstructions

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Flow matching and flow sampling · 0 citations (OpenAlex)

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Studies when stochastic processes with independent endpoints induce straight-line ODE flows; gives PDE characterizations, explicit Gaussian constructions, and impossibility theorems for well-separated multimodal targets.

Problem

Which stochastic processes X with independent endpoints (X_0,X_1) ~ P_0 ⊗ P_1 induce straight-line (zero-acceleration) velocity flows for generative transport, enabling single-step integration?

Approach

Derives Eulerian PDE characterizations of straightness via the Reynolds stress and material derivative; constructs a generalized interpolant with a Gaussian correction term that yields straight flows for arbitrary non-degenerate Gaussian endpoints; proves that under endpoint independence, straight-line processes cannot exist when the target has well-separated modes, using space-time topological no

Key results

• Affine processes X_t=(1-t)X_0+tX_1 induce straight flows only if the endpoint coupling is deterministic.
• For arbitrary non-degenerate Gaussian endpoints, the interpolant F(t,x,y,z)=(1-t)x+ty+sqrt(2t(1-t))z with Gaussian Z yields a straight-line process in F_SL.
• For sources P_0 and targets P_1 with properly disconnected support in d=1 (and P_0≠P_1 weights in higher d), S_SL(P,P) is empty.

Surprising findings

• Straight-line processes with independent endpoints can be impossible even for a Gaussian source and a multimodal Gaussian-mixture target, contrary to the expectation that such constructions are always available.
• The constant process X_t = X_0 does not give an independent-endpoint straight process for P,P, since its endpoints are identical.

Limitations

• Impossibility for d>1 with equal mode weights and connected-support cases is not fully covered.
• Results are theoretical with no empirical experiments on generative models.
• Gaussian construction is restricted to Gaussian endpoints; generic unimodal targets are left open.

Experimental setup

Builds on: stochastic interpolants rectified flow flow matching Brownian bridge Monge optimal transport

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