Diffusion model is an emerging generative modeling technique, achieving the state-of-the-art performances in image and video synthesis, scientific simulation, inverse problems, and offline reinforcement learning. Yet, existing statistical analysis of diffusion models often requires restrictive theoretical assumptions or is suboptimal. In this talk, we present two recent works from our group toward closing these gaps between diffusion models and the theoretical limits in standard nonparametric and high-dimensional statistical settings, and discuss some future directions.

1) For subGaussian distributions on $\mathbb{R}^d$ with $\beta$-Hölder smooth densities ($\beta\le 2$), we show that the sampling distribution of diffusion models can be minimax optimal under the total variation distance, even if the score is learned from noise perturbed training samples with noise multiplicity equal to one and without density lower bound assumptions.

2) For conditional sampling under i.i.d. priors and noisy linear observations, we show that diffusion models (also known as stochastic localization) can successfully sample from the posterior distribution, provided the signal-to-noise ratio exceeds a computational threshold predicted by prior work on approximate message passing (Barbier et al., 2020). This improves previous thresholds established in the stochastic localization literature, and enhances the sampling accuracy of dominant noisy inverse sampling techniques used in machine learning - albeit in a stylized theoretical model.

(Based on works with Kaihong Zhang, Heqi Yin, Feng Liang arXiv: 2402.15602, and with Han Cui, and Zhiyuan Yu arXiv: 2407.10763)