# The Principles of Diffusion Models

arXiv: https://arxiv.org/abs/2510.21890 · HTML: https://mlsanity.com/paper/2510.21890/

## Key facts

- Cited by 76 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- First posted on arXiv 24 Oct 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Built on by (most cited first, 30 of 76)

- [A Unified View of Score-Based and Drifting Models](https://mlsanity.com/paper/2603.07514/) (2026)
- [From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity](https://mlsanity.com/paper/2512.02826/) (2025)
- [Towards a unified framework for guided diffusion models](https://mlsanity.com/paper/2512.04985/) (2025)
- [Noise Scheduling as Information-Guided Allocation in Diffusion Training](https://mlsanity.com/paper/2602.18647/) (2026)
- [Compositional Planning with Jumpy World Models](https://mlsanity.com/paper/2602.19634/) (2026)
- [SLICE: Semantic Latent Injection via Compartmentalized Embedding for Image Watermarking](https://mlsanity.com/paper/2603.12749/) (2026)
- [V-GRPO: Online Reinforcement Learning for Denoising Generative Models Is Easier than You Think](https://mlsanity.com/paper/2604.23380/) (2026)
- [MeanFlow Transformers with Representation Autoencoders](https://mlsanity.com/paper/2511.13019/) (2025)
- [OmniGuide: Universal Guidance Fields for Enhancing Generalist Robot Policies](https://mlsanity.com/paper/2603.10052/) (2026)
- [UniSteer: Unified Noise Steering for Efficient Human-Guided VLA Adaptation](https://mlsanity.com/paper/2605.10821/) (2026)
- [Dynamical Regimes of Multimodal Diffusion Models](https://mlsanity.com/paper/2602.04780/) (2026)
- [Conditioning Gaussian Processes on Almost Anything](https://mlsanity.com/paper/2605.21041/) (2026)
- [Provable diffusion-based posterior sampling for linear inverse problems via DDIM](https://mlsanity.com/paper/2607.19333/) (2026)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Closed-form conditional diffusion models for data assimilation](https://mlsanity.com/paper/2603.21291/) (2026)
- [Generalized Discrete Diffusion from Snapshots](https://mlsanity.com/paper/2603.21342/) (2026)
- [One Step Forward and K Steps Back: Better Reasoning with Denoising Recursion Models](https://mlsanity.com/paper/2604.18839/) (2026)
- [Scalable Differentially Private Data Compression via Diffusion and Stochastic Codes](https://mlsanity.com/paper/2607.03392/) (2026)
- [On The Hidden Biases of Flow Matching Samplers](https://mlsanity.com/paper/2512.16768/) (2025)
- [EditYourself: Audio-Driven Generation and Manipulation of Talking Head Videos with Diffusion Transformers](https://mlsanity.com/paper/2601.22127/) (2026)
- [Stabilizing Physics-Informed Consistency Models via Structure-Preserving Training](https://mlsanity.com/paper/2602.09303/) (2026)
- [Tucker Diffusion Model for High-dimensional Tensor Generation](https://mlsanity.com/paper/2604.00481/) (2026)
- [Unlearning in Diffusion Models: A Unified Framework with KL Divergence and Likelihood Constraints](https://mlsanity.com/paper/2605.30825/) (2026)
- [Accelerating Diffusion Sampling via Speculative Draft Trees](https://mlsanity.com/paper/2609.17691/) (2026)
- [Generative AI Meets 6G and Beyond: Diffusion Models for Semantic Communications](https://mlsanity.com/paper/2511.08416/) (2025)
- [Iterative Sampling Methods for Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2512.12550/) (2025)
- [Residual Prior Diffusion: A Probabilistic Framework Integrating Coarse Latent Priors with Diffusion Models](https://mlsanity.com/paper/2512.21593/) (2025)
- [Direct Soft-Policy Sampling via Langevin Dynamics](https://mlsanity.com/paper/2602.07873/) (2026)
- [CMAD: Cooperative Multi-Agent Diffusion via Stochastic Optimal Control](https://mlsanity.com/paper/2602.10933/) (2026)
- [MGD: Moment Guided Diffusion for Maximum Entropy Generation](https://mlsanity.com/paper/2602.17211/) (2026)

## Abstract (arXiv)

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/