# Geometry of Score Based Generative Models

arXiv: https://arxiv.org/abs/2302.04411 · HTML: https://mlsanity.com/paper/2302.04411/

## Key facts

- Cited by 7 papers in the map (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 Feb 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)

- [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)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Flow Contrastive Estimation of Energy-Based Models](https://mlsanity.com/paper/1912.00589/) (2019)
- [Unsupervised Learning of Compositional Energy Concepts](https://mlsanity.com/paper/2111.03042/) (2021)
- [A Non-Asymptotic Analysis for Stein Variational Gradient Descent](https://mlsanity.com/paper/2006.09797/) (2020)
- [WaveFit: An Iterative and Non-autoregressive Neural Vocoder based on Fixed-Point Iteration](https://mlsanity.com/paper/2210.01029/) (2022)

## Built on by (most cited first, 7 of 7)

- [Towards Faster Non-Asymptotic Convergence for Diffusion-Based Generative Models](https://mlsanity.com/paper/2306.09251/) (2023)
- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Correlated and Multi-frequency Diffusion Modeling for Highly Under-sampled MRI Reconstruction](https://mlsanity.com/paper/2309.00853/) (2023)
- [The Spacetime of Diffusion Models: An Information Geometry Perspective](https://mlsanity.com/paper/2505.17517/) (2025)
- [Are We Really Learning the Score Function? Reinterpreting Diffusion Models Through Wasserstein Gradient Flow Matching](https://mlsanity.com/paper/2509.00336/) (2025)
- [Deep Generative Modeling with Backward Stochastic Differential Equations](https://mlsanity.com/paper/2304.04049/) (2023)
- [Resistive Memory-based Neural Differential Equation Solver for Score-based Diffusion Model](https://mlsanity.com/paper/2404.05648/) (2024)

## Abstract (arXiv)

In this work, we look at Score-based generative models (also called diffusion generative models) from a geometric perspective. From a new view point, we prove that both the forward and backward process of adding noise and generating from noise are Wasserstein gradient flow in the space of probability measures. We are the first to prove this connection. Our understanding of Score-based (and Diffusion) generative models have matured and become more complete by drawing ideas from different fields like Bayesian inference, control theory, stochastic differential equation and Schrodinger bridge. However, many open questions and challenges remain. One problem, for example, is how to decrease the sampling time? We demonstrate that looking from geometric perspective enables us to answer many of these questions and provide new interpretations to some known results. Furthermore, geometric perspective enables us to devise an intuitive geometric solution to the problem of faster sampling. By augmenting traditional score-based generative models with a projection step, we show that we can generate high quality images with significantly fewer sampling-steps.