The Principles of Diffusion Models
cs.LG, cs.AI, cs.GR
Submitted: 2025-10-24
Updated: 2026-08-27
Comments: Supplementary materials for the book are available at the book website: https://the-principles-of-diffusion-models.github.io/
Project page: https://the-principles-of-diffusion-models.github.io
License: http://creativecommons.org/licenses/by-nc-nd/4.0/
The gist: 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.
Terminology
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. Supplementary materials for the book are available at the book website: https://the-principles-of-diffusion-models.github.io/
Sources
- Stochastic Interpolants: A Unifying Framework for Flows and Diffusions
- Diffusion Posterior Sampling for General Noisy Inverse Problems
- A Survey on Diffusion Models for Inverse Problems
- Convergence of denoising diffusion models under the manifold hypothesis
- Generative Modeling via Drifting
- Continuous diffusion for categorical data
- Discrete Flow Matching
- Consistency Models Made Easy
- On the Relation between Rectified Flows and Optimal Transport
- CMT: Mid-Training for Efficient Learning of Consistency, Mean Flow, and Flow Map Models
- Analyzing and Improving the Training Dynamics of Diffusion Models
- PaGoDA: Progressive Growing of a One-Step Generator from a Low-Resolution Diffusion Teacher
- Auto-Encoding Variational Bayes
- A Unified View of Score-Based and Drifting Models
- On the Equivalence of Consistency-Type Models: Consistency Models, Consistent Diffusion Models, and Fokker-Planck Regularization
- Flow Matching Guide and Code
- Rectified Flow: A Marginal Preserving Approach to Optimal Transport
- Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models
- DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models
- Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed
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