# Generative Diffusion From An Action Principle

arXiv: https://arxiv.org/abs/2310.04490 · HTML: https://mlsanity.com/paper/2310.04490/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 6 Oct 2023.

## Placed in

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

## Builds on (most cited first, 14 of 14)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Variational Walkback: Learning a Transition Operator as a Stochastic Recurrent Net](https://mlsanity.com/paper/1711.02282/) (2017)
- [On local entropy, stochastic control and deep neural networks](https://mlsanity.com/paper/2204.13049/) (2022)

## Built on by (most cited first, 5 of 5)

- [Understanding Diffusion Models by Feynman's Path Integral](https://mlsanity.com/paper/2403.11262/) (2024)
- [Neural Entropy](https://mlsanity.com/paper/2409.03817/) (2024)
- [Diffusion Density Estimators](https://mlsanity.com/paper/2410.06986/) (2024)
- [A Unified Kullback--Leibler Divergence Analysis of Generative Diffusion Models via Entropy Production Rate](https://mlsanity.com/paper/2608.02406/) (2026)
- [The Loss Floor of Denoising Score Matching: Fisher Geometry from Schr"odinger Bridges](https://mlsanity.com/paper/2608.23916/) (2026)

## Abstract (arXiv)

Generative diffusion models synthesize new samples by reversing a diffusive process that converts a given data set to generic noise. This is accomplished by training a neural network to match the gradient of the log of the probability distribution of a given data set, also called the score. By casting reverse diffusion as an optimal control problem, we show that score matching can be derived from an action principle, like the ones commonly used in physics. We use this insight to demonstrate the connection between different classes of diffusion models.