The Principles of Diffusion Models

2025/10/24 by Chieh-Hsin Lai, Yang Song, Lai, Chieh-Hsin +7 · 21 voices · 10 citations
#cs.LG #cs.AI #cs.GR

paper · pdf · doi:10.48550/arxiv.2510.21890

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.

Cited by

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• Conditioning Gaussian Processes on Almost Anything
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• Residual Prior Diffusion: A Probabilistic Framework Integrating Coarse Latent Priors with Diffusion Models
• Iterative Sampling Methods for Sinkhorn Distributionally Robust Optimization
• From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity
• MeanFlow Transformers with Representation Autoencoders
• Align & Invert: Solving Inverse Problems with Diffusion and Flow-based Models via Representation Alignment
• Generative AI Meets 6G and Beyond: Diffusion Models for Semantic Communications
• CMT: Mid-Training for Efficient Learning of Consistency, Mean Flow, and Flow Map Models
• A Unifying Framework for Parallelizing Sequential Models with Linear Dynamical Systems

Discussions

• The Principles of Diffusion Models [hn, 239 points, 31 comments]
• "The Principles of Diffusion Models" by Chieh-Hsin Lai, Yang Song, Dongjun Kim, Yuki Mitsufuji, Stefano Ermon. arxiv.org/abs/2510.21890 It might not be the easiest intro to diffusion models, but this [bsky, 37 points, 1 comments]
• babe! it's 4pm, time for your 400 pages of diffusion model theory [bsky, 25 points, 2 comments]
• The Principles of Diffusion Models [hn, 10 points, 1 comments]
• The Principles of Diffusion Models arxiv.org/abs/2510.21890 Discuss: www.sqox.com/c/?id=694394... [bsky, 1 points, 0 comments]
• 拡散モデルのメモ: www.arxiv.org/abs/2510.21890 [bsky, 1 points, 0 comments]
• The Principles of Diffusion Models https://arxiv.org/abs/2510.21890 https://news.ycombinator.com/item?id=45866572 [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models https://arxiv.org/abs/2510.21890 [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models View Article | Join the HN Conversation Summary of HN discussion 🧵👇 #hacker-news [bsky, 0 points, 1 comments]
• The Principles of Diffusion Models https://arxiv.org/abs/2510.21890 [comments] [124 points] [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models https://arxiv.org/abs/2510.21890 (https://news.ycombinator.com/item?id=45866572) [bsky, 0 points, 0 comments]
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• The Principles of Diffusion Models https://arxiv.org/abs/2510.21890 (http://news.ycombinator.com/item?id=45866572) [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models arxiv.org/abs/2510.21890 470 page monograph from OpenAI, Sony AI, Stanford. [bsky, 0 points, 0 comments]
• https://arxiv.org/abs/2510.21890 拡散モデルの原理に関する論文です。 数式と解説で詳細に原理を説明しています。 AI技術に興味がある人におすすめです。 [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models https:// arxiv.org/abs/2510.21890 # HackerNews # DiffusionModels # Principles # AI # Research # MachineLearning # Arxiv [mastodon, 0 points, 0 comments]
• https://bsky.app/profile/buzzing.cc.web.brid.gy/post/3m5a7vl5szrw2 [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models https://arxiv.org/abs/2510.21890 (https://news.ycombinator.com/item?id=45866572) [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models arxiv.org/pdf/2510.21890 [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models [bsky, 0 points, 0 comments]
• The Principles of Diffusion Models #HackerNews https://arxiv.org/abs/2510.21890 [bsky, 0 points, 0 comments]

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