Linear Convergence of Diffusion Models Under the Manifold Hypothesis

Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into $D$-dimensional space; this is known as the manifold hypothesis. The current best-known convergence guarantees are either linear in $D$ or polynomial (superlinear) in $d$. The latter exploits a novel integration scheme for the backward SDE. We take the best of both worlds and show that the number of steps diffusion models require in order to converge in Kullback-Leibler~(KL) divergence is linear (up to logarithmic terms) in the intrinsic dimension $d$. Moreover, we show that this linear dependency is sharp.

@article{potaptchik2025_2410.09046,
  title={ Linear Convergence of Diffusion Models Under the Manifold Hypothesis },
  author={ Peter Potaptchik and Iskander Azangulov and George Deligiannidis },
  journal={arXiv preprint arXiv:2410.09046},
  year={ 2025 }
}