# Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions

arXiv: https://arxiv.org/abs/2507.07008 · HTML: https://mlsanity.com/paper/2507.07008/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 27 of its 42 references that are in the map.
- First posted on arXiv 9 Jul 2025.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 27 of 27)

- [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)
- [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)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [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)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems](https://mlsanity.com/paper/2503.08136/) (2025)
- [Learning Diffusion Priors from Observations by Expectation Maximization](https://mlsanity.com/paper/2405.13712/) (2024)
- [An analysis of the noise schedule for score-based generative models](https://mlsanity.com/paper/2402.04650/) (2024)
- [Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors](https://mlsanity.com/paper/2405.14250/) (2024)
- [From Score Matching to Diffusion: A Fine-Grained Error Analysis in the Gaussian Setting](https://mlsanity.com/paper/2503.11615/) (2025)
- [Do Bayesian imaging methods report trustworthy probabilities?](https://mlsanity.com/paper/2405.08179/) (2024)
- [Evaluating the Posterior Sampling Ability of Plug&Play Diffusion Methods in Sparse-View CT](https://mlsanity.com/paper/2410.21301/) (2024)
- [Stochastic super-resolution for Gaussian microtextures](https://mlsanity.com/paper/2405.15399/) (2024)

## Abstract (arXiv)

Used as priors for Bayesian inverse problems, diffusion models have recently attracted considerable attention in the literature. Their flexibility and high variance enable them to generate multiple solutions for a given task, such as inpainting, super-resolution, and deblurring. However, there is still a lack of understanding about how accurately these conditional diffusion algorithms perform conditional sampling. In this article, we investigate the errors induced by these models when applied to a Gaussian data distribution for which the score function is exactly known. Within this constrained context, we are able to precisely analyze the discrepancy between the theoretical resolution of inverse problems via conditional sampling and the practical distributions generated by conditional diffusion models. This is done by characterizing all the involved iterative Gaussian processes and by computing the exact Wasserstein distance between the distributions of the diffusion model samplers and the ideal conditional distribution associated with the inverse problem. Our findings allow for the comparison of two major algorithms from the literature, Deep Posterior Sampling (DPS) and Pseudo-inverse Guided Diffusion Models (PiGDM), and the introduction of the new paradigm Conditional Gaussian Diffusion Models (CGDM) that is shown to be more accurate for Gaussian data distributions.