Generative diffusion models are one of the most successful and flexible families of AI tools, with applications today going far beyond their original goal of sampling from an empirically learned image distribution. They are also one of the few modern AI models that originated from well-established mathematical ideas—namely, the theory of Markov diffusion processes and their time-reversals. Despite this early connection, and many subsequent developments drawing inspiration from probabilistic ideas in case-specific ways, the theory of generative diffusion models as probabilistic objects themselves is surprisingly far from complete.
We will first overview the basic ideas of generative diffusion models from a machine learning practitioner’s viewpoint. Then, we will propose a purely probabilistic perspective on these models, their aims, and their training methods, leveraging stochastic analysis tools and results—some of them classic, such as Girsanov’s theorem and Doob’s h-transform, and some less standard, like Föllmer processes and the enlargement of filtrations. We will also discuss how these concepts are (explicitly or not) built-in and utilized by goal-specific sampling methods, such as classifier-free guidance and Schrödinger Bridge matching.
On this basis, we will propose a general probabilistic framework that unifies several generative diffusion settings, their sampling goals, and their training methods. If time permits, we will also discuss some potential new (yet to be developed) applications of this framework, including in multimodal conditional sampling (e.g., text-to-image and vice versa).
 Based on ongoing works with David Felipe and Lucas Villanueva (U. of Chile)