Diffusion models have recently gained attention as a powerful class of deep generative models, achieving state-of-the-art results in data generation tasks. In a nutshell, they are designed to learn an unknown data distribution starting from Gaussian noise, mimicking the process of non-equilibrium thermodynamic diffusion. Despite their outstanding empirical successes, the mathematical and algorithmic foundations of diffusion models remain far from mature. For instance: (i) Generalization: it remains unclear how diffusion models, trained on finite samples, can generate new and meaningful data that differ from the training set; (ii) Efficiency: due to the enormous model capacity and the requirement of many sampling steps, they often suffer from slow training and sampling speeds; (iii) Controllability: it remains computationally challenging and unclear how to guide and control the content generated by diffusion models, raising challenges over controllability and safety, as well as solving inverse problems across many scientific imaging applications.

This tutorial will introduce a mathematical framework for understanding the generalization and improving the efficiency of diffusion models, through exploring the low-dimensional structures in both the data and model. We show how to overcome fundamental barriers to improve the generalization, efficiency, and controllability in developing diffusion models, by exploring how these models adaptively learn underlying data distributions, how to achieve faster convergence at the sampling stage, and unveiling the intrinsic properties of the learned denoiser. Leveraging the theoretical studies, we will show how to effectively employ these properties for controlling the generation of diffusion models.