🤖 AI Summary
This work addresses the challenge faced by beginning graduate students who lack prior exposure to stochastic differential equations and diffusion models by proposing a hierarchical pedagogical framework that systematically constructs the mathematical foundations of diffusion models. Starting from a sampling perspective, it integrates core definitions, key estimates under simplified assumptions, and proof strategies for cutting-edge theorems, thereby bridging classical sampling dynamics with modern diffusion samplers. The material synthesizes probability theory, stochastic differential equations, stochastic numerical methods, and diffusion process theory into a self-contained, proof-oriented curriculum. This approach maintains mathematical rigor while significantly enhancing accessibility, enabling students without prerequisite knowledge to grasp the sampling mechanisms, error analysis, and inference control principles underlying diffusion models.
📝 Abstract
These notes give a proof-oriented introduction to diffusion models from the viewpoint of sampling, tracing a single arc from classical sampling dynamics to modern diffusion samplers, their error analysis, and inference-time control. Throughout, the material is layered into core definitions and identities proved in full, representative estimates proved under simplifying assumptions, and research-level theorems stated with a proof roadmap. The intended audience is beginning graduate students with a background in probability but no prior exposure to stochastic differential equations, stochastic numerics, or diffusion models.