Math paper tightens the theory behind diffusion samplers and PF-ODEs
chaumian · x · 2026-07-29
Key points
This paper studies the mathematical relationship between diffusion processes, the Fokker–Planck equation, and the probability-flow ODE (PF-ODE) used in score-based generative modeling.
- It proves existence and uniqueness of the marginal density flow under minimal assumptions on diffusion coefficients.
- Using regular Lagrangian flow theory, it shows the PF-ODE transports the initial distribution to the diffusion marginals under Sobolev or bounded-variation regularity of the score, plus one-sided divergence bounds.
- The paper also identifies sufficient regularity conditions relevant to diffusion models used in practice.
- A counterexample shows that the Fokker–Planck equation can have a unique density flow while the associated PF-ODE still fails, highlighting a real Eulerian/Lagrangian mismatch.
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