The Principles of Diffusion Models
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.
6901dcd6646208eac0d1f5d0
Chieh-Hsin Lai
6901dcd6646208eac0d1f5d1
Yang Song
6901dcd6646208eac0d1f5d2
Dongjun Kim
6901dcd6646208eac0d1f5d3
Yuki Mitsufuji
6901dcd6646208eac0d1f5d4
Stefano Ermon
The Principles of Diffusion Models
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.
Diffusion models are explored through variational, score-based, and flow-based perspectives, focusing on their mathematical foundations and applications in controllable generation and efficient sampling.
diffusion models variational view variational autoencoders score-based view energy-based modeling flow-based view normalizing flows time-dependent velocity field differential equation controllable generation efficient numerical solvers diffusion-motivated flow-map models