# FreeFlow: A Comprehensive Understanding on Diffusion Probabilistic Models via Optimal Transport

arXiv: https://arxiv.org/abs/2312.05486 · HTML: https://mlsanity.com/paper/2312.05486/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 20 of its 27 references that are in the map.
- First posted on arXiv 9 Dec 2023.

## Placed in

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

## Builds on (most cited first, 20 of 20)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (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)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [Poisson Flow Generative Models](https://mlsanity.com/paper/2209.11178/) (2022)
- [GenPhys: From Physical Processes to Generative Models](https://mlsanity.com/paper/2304.02637/) (2023)
- [Text-Driven Stylization of Video Objects](https://mlsanity.com/paper/2206.12396/) (2022)
- [PaddleSpeech: An Easy-to-Use All-in-One Speech Toolkit](https://mlsanity.com/paper/2205.12007/) (2022)
- [Faster Training of Diffusion Models and Improved Density Estimation via Parallel Score Matching](https://mlsanity.com/paper/2306.02658/) (2023)

## Abstract (arXiv)

The blooming diffusion probabilistic models (DPMs) have garnered significant interest due to their impressive performance and the elegant inspiration they draw from physics. While earlier DPMs relied upon the Markovian assumption, recent methods based on differential equations have been rapidly applied to enhance the efficiency and capabilities of these models. However, a theoretical interpretation encapsulating these diverse algorithms is insufficient yet pressingly required to guide further development of DPMs. In response to this need, we present FreeFlow, a framework that provides a thorough explanation of the diffusion formula as time-dependent optimal transport, where the evolutionary pattern of probability density is given by the gradient flows of a functional defined in Wasserstein space. Crucially, our framework necessitates a unified description that not only clarifies the subtle mechanism of DPMs but also indicates the roots of some defects through creative involvement of Lagrangian and Eulerian views to understand the evolution of probability flow. We particularly demonstrate that the core equation of FreeFlow condenses all stochastic and deterministic DPMs into a single case, showcasing the expansibility of our method. Furthermore, the Riemannian geometry employed in our work has the potential to bridge broader subjects in mathematics, which enable the involvement of more profound tools for the establishment of more outstanding and generalized models in the future.