Table of Contents
About this Web Edition

This website is an unofficial, responsive digital version converted from the open-access monograph manuscript The Principles of Diffusion Models (arXiv:2510.21890) by Chieh-Hsin Lai et al. All original content and copyright belong to the respective authors and publisher.

FORTHCOMING FROM MIT PRESS, 2027

The Principles of Diffusion Models

Chieh-Hsin Lai, Yang Song, Dongjun Kim, Yuki Mitsufuji, and Stefano Ermon
Start Reading (Preface and Roadmap) →

About This Book

This book focuses on the principles that have shaped the development of diffusion models, tracing their origins and showing how different formulations arise from common mathematical ideas.

Diffusion modeling begins by specifying a forward corruption process that gradually turns data into noise. This forward process links the data distribution to a simple noise distribution by defining a continuous family of intermediate distributions. The core objective of a diffusion model is to construct another process that runs in the opposite direction, transforming noise into data while recovering the same intermediate distributions defined by the forward corruption process.

Table of Contents

Front Matter

Front Matter
Acknowledgements
Front Matter
Preface and Roadmap
Front Matter
Notations

Part A: Introduction to Deep Generative Modeling

Part A: Introduction to Deep Generative Modeling
Chapter 1: Deep Generative Modeling

Part B: Foundations and Perspectives of Diffusion Models

Part B: Foundations and Perspectives of Diffusion Models
Part B Overview & Diagram
Part B: Foundations and Perspectives of Diffusion Models
A First Operational View of Diffusion Model Training and Sampling
Part B: Foundations and Perspectives of Diffusion Models
Chapter 2: Variational Perspective: From VAEs to DDPMs
Part B: Foundations and Perspectives of Diffusion Models
Chapter 3: Score-Based Perspective: From EBMs to NCSN
Part B: Foundations and Perspectives of Diffusion Models
Chapter 4: Diffusion Models Today: Score SDE Framework
Part B: Foundations and Perspectives of Diffusion Models
Chapter 5: Flow-Based Perspective: From NFs to Flow Matching
Part B: Foundations and Perspectives of Diffusion Models
Chapter 6: A Unified and Systematic Lens on Diffusion Models
Part B: Foundations and Perspectives of Diffusion Models
Chapter 7: (Optional) Diffusion Models and Optimal Transport

Part C: Sampling of Diffusion Models

Part C: Sampling of Diffusion Models
Part C & D Overview & Diagram
Part C: Sampling of Diffusion Models
Chapter 8: Guidance and Controllable Generation
Part C: Sampling of Diffusion Models
Chapter 9: Numerical Solvers for Fast Diffusion Sampling

Part D: Toward Learning Fast Diffusion-Based Generators

Part D: Toward Learning Fast Diffusion-Based Generators
Chapter 10: Distillation-Based Methods for Fast Sampling
Part D: Toward Learning Fast Diffusion-Based Generators
Chapter 11: Learning Fast Generators from Scratch

Epilogue & Outlook

Epilogue & Outlook
Chapter 12: A Unifying Principle and the Road Ahead

Appendices

Appendices
Appendix A: Crash Course on Differential Equations
Appendices
Appendix B: Density Evolution: From Change of Variable to Fokker–Planck
Appendices
Appendix C: Behind the Scenes of Diffusion Models: Itô’s Calculus and Girsanov’s Theorem
Appendices
Appendix D: Supplementary Materials and Proofs