The Principles of Diffusion Models
Authors: name: Chieh-Hsin Lai, name: Yang Song, name: Dongjun Kim, name: Yuki Mitsufuji, name: Stefano Ermon

This monograph presents the core principles that have guided the development
of diffusion models, tracing their origins and showing how diverse formulations
arise from shared mathematical ideas. Diffusion modeling starts by defining a
forward process that gradually corrupts data into noise, linking the data
distribution to a simple prior through a continuum of intermediate
distributions. The goal is to learn a reverse process that transforms noise
back into data while recovering the same intermediates. We describe three
complementary views. The variational view, inspired by variational
autoencoders, sees diffusion as learning to remove noise step by step. The
score-based view, rooted in energy-based modeling, learns the gradient of the
evolving data distribution, indicating how to nudge samples toward more likely
regions. The flow-based view, related to normalizing flows, treats generation
as following a smooth path that moves samples from noise to data under a
learned velocity field. These perspectives share a common backbone: a
time-dependent velocity field whose flow transports a simple prior to the data.
Sampling then amounts to solving a differential equation that evolves noise
into data along a continuous trajectory. On this foundation, the monograph
discusses guidance for controllable generation, efficient numerical solvers,
and diffusion-motivated flow-map models that learn direct mappings between
arbitrary times. It provides a conceptual and mathematically grounded
understanding of diffusion models for readers with basic deep-learning
knowledge.