Diffusion Models are Minimax Optimal Distribution Estimators

We provide the first rigorous analysis on estimation error bounds of diffusion modeling for well-known function spaces. The highlight of this paper is that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates in the total variation distance and in the Wasserstein distance of order one. We expect these results advance theoretical understandings of diffusion modeling and its ability to generate verisimilar outputs.