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A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models
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abstract
Diffusion models, which convert noise into new data instances by learning to reverse a diffusion process, have become a cornerstone in contemporary generative modeling. In this work, we develop non-asymptotic convergence theory for a popular diffusion-based sampler (i.e., the probability flow ODE sampler) in discrete time, assuming access to $\ell_2$-accurate estimates of the (Stein) score functions. For distributions in $\mathbb{R}^d$, we prove that $d/\varepsilon$ iterations -- modulo some logarithmic and lower-order terms -- are sufficient to approximate the target distribution to within $\varepsilon$ total-variation distance. This is the first result establishing nearly linear dimension-dependency (in $d$) for the probability flow ODE sampler. Imposing only minimal assumptions on the target data distribution (e.g., no smoothness assumption is imposed), our results also characterize how $\ell_2$ score estimation errors affect the quality of the data generation processes. In contrast to prior works, our theory is developed based on an elementary yet versatile non-asymptotic approach without the need of resorting to SDE and ODE toolboxes.
Forward citations
Cited by 6 Pith papers
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A Sharp KL-Convergence Analysis for Diffusion Models under Minimal Assumptions
A new analysis shows O~(d/epsilon) steps suffice for KL-close diffusion sampling under only L2 score error and finite second moment assumptions, improving the known O~(d/epsilon^2).
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Inexact generative models stay on the data manifold because infinitesimal learning errors perturb the density only along the manifold, when top Lyapunov vectors align with the support boundary.
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Generalization bounds for score-based generative models: a synthetic proof
Score-based generative models achieve minimax optimal W1 rates n^{-(β+1)/(2β+d)} over β-Hölder densities, up to polylog factors.
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Faster Diffusion Models via Higher-Order Approximation
A new higher-order ODE sampler for diffusion models is proven to reach ε total-variation accuracy with eO(d^{1+2/K}/ε^{1/K}) iterations under mild assumptions.
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Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models
A TV convergence bound O(d^{7/4} ε^{1/2} + d(dH)^p) is proved for p-th order (exponential) Runge-Kutta samplers of probability-flow ODEs under C² smoothness of the learned score.
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Non-asymptotic convergence bound of conditional diffusion models
CARD's generated conditional distribution is shown to converge in Wasserstein distance to the true conditional distribution, with a separate score-estimation error bound controlled by network resolution and distributi...
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