A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models
<|reference_start|>A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models: 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.<|reference_end|>
@article{li2024a,
    title={A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion
  Models},
    author={Gen Li and Yuting Wei and Yuejie Chi and Yuxin Chen},
    journal={arXiv preprint arXiv:2408.02320},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.02320},
    primaryClass={cs.LG cs.NA eess.SP math.NA math.ST stat.ML stat.TH}
}