Paper 2510.21890
The Principles of Diffusion Models
- Published
- Oct 2025
- Research lab
- Independent
- Citations
- 69
- GitHub
- Not linked
01 In brief
Summary
This book provides a comprehensive, mathematically grounded introduction to diffusion models, tracing their origins from three foundational perspectives: variational (VAEs to DDPMs), score-based (EBMs to NCSN and Score SDEs), and flow-based (Normalizing Flows to Flow Matching).
It unifies these views through the Fokker-Planck equation and a common conditioning trick, showing that different parameterizations (noise, clean, score, velocity) are equivalent.
The book then covers practical aspects: guidance for controllable generation (classifier, classifier-free, training-free), advanced numerical solvers for fast sampling (DDIM, DEIS, DPM-Solver), and training-based acceleration via distillation and learning flow-map models from scratch (Consistency Models, CTM, Mean Flow).
It also explores connections to optimal transport and extends the core principles to discrete state spaces.
The book is intended for readers with a basic deep learning background and serves as a principled reference for researchers and students.
02 From the paper
Abstract
This book 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 book 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.