# Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors

arXiv: https://arxiv.org/abs/2405.14250 · HTML: https://mlsanity.com/paper/2405.14250/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 24 of its 35 references that are in the map.
- First posted on arXiv 23 May 2024.

## Placed in

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

## Builds on (most cited first, 24 of 24)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [A Wasserstein-type distance in the space of Gaussian Mixture Models](https://mlsanity.com/paper/1907.05254/) (2019)
- [Learning Mixtures of Gaussians Using the DDPM Objective](https://mlsanity.com/paper/2307.01178/) (2023)
- [Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distances](https://mlsanity.com/paper/2401.17958/) (2024)
- [Understanding DDPM Latent Codes Through Optimal Transport](https://mlsanity.com/paper/2202.07477/) (2022)
- [How Much is Enough? A Study on Diffusion Times in Score-based Generative Models](https://mlsanity.com/paper/2206.05173/) (2022)
- [Explicit Diffusion of Gaussian Mixture Model Based Image Priors](https://mlsanity.com/paper/2302.08411/) (2023)
- [Product of Gaussian Mixture Diffusion Models](https://mlsanity.com/paper/2310.12653/) (2023)

## Built on by (most cited first, 16 of 16)

- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [An Analytical Theory of Spectral Bias in the Learning Dynamics of Diffusion Models](https://mlsanity.com/paper/2503.03206/) (2025)
- [Spectral Analysis of Diffusion Models with Application to Schedule Design](https://mlsanity.com/paper/2502.00180/) (2025)
- [Wasserstein Bounds for generative diffusion models with Gaussian tail targets](https://mlsanity.com/paper/2412.11251/) (2024)
- [Diffusion and Flow-based Copulas: Forgetting and Remembering Dependencies](https://mlsanity.com/paper/2509.19707/) (2025)
- [A Random Matrix Theory Perspective on the Consistency of Diffusion Models](https://mlsanity.com/paper/2602.02908/) (2026)
- [A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm](https://mlsanity.com/paper/2509.20511/) (2025)
- [Geometry-Aware Discretization Error of Diffusion Models](https://mlsanity.com/paper/2605.08392/) (2026)
- [Optimal Stopping in Latent Diffusion Models](https://mlsanity.com/paper/2510.08409/) (2025)
- [When Are Two Scores Better Than One? Investigating Ensembles of Diffusion Models](https://mlsanity.com/paper/2601.11444/) (2026)
- [Alternating Diffusion for Proximal Sampling with Zeroth Order Queries](https://mlsanity.com/paper/2603.19633/) (2026)
- [The critical slowing down in training diffusion models](https://mlsanity.com/paper/2605.12597/) (2026)
- [Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions](https://mlsanity.com/paper/2507.07008/) (2025)
- [A Gaussian Perspective for Distributional Discrepancy in Generative Diffusion Models](https://mlsanity.com/paper/2601.13602/) (2026)
- [A Unified Kullback--Leibler Divergence Analysis of Generative Diffusion Models via Entropy Production Rate](https://mlsanity.com/paper/2608.02406/) (2026)
- [A Stochastic Analysis Perspective on Diffusion Models](https://mlsanity.com/paper/2610.14133/) (2026)

## Abstract (arXiv)

Diffusion or score-based models recently showed high performance in image generation. They rely on a forward and a backward stochastic differential equations (SDE). The sampling of a data distribution is achieved by numerically solving the backward SDE or its associated flow ODE. Studying the convergence of these models necessitates to control four different types of error: the initialization error, the truncation error, the discretization error and the score approximation. In this paper, we theoretically study the behavior of diffusion models and their numerical implementation when the data distribution is Gaussian. Our first contribution is to derive the analytical solutions of the backward SDE and the probability flow ODE and to prove that these solutions and their discretizations are all Gaussian processes. Our second contribution is to compute the exact Wasserstein errors between the target and the numerically sampled distributions for any numerical scheme. This allows us to monitor convergence directly in the data space, while experimental works limit their empirical analysis to Inception features. An implementation of our code is available online.