# Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data

arXiv: https://arxiv.org/abs/2403.08728 · HTML: https://mlsanity.com/paper/2403.08728/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 41 of its 59 references that are in the map.
- First posted on arXiv 13 Mar 2024.

## Placed in

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

## Builds on (most cited first, 30 of 41)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Diffusion models as plug-and-play priors](https://mlsanity.com/paper/2206.09012/) (2022)
- [Fast Training of Diffusion Models with Masked Transformers](https://mlsanity.com/paper/2306.09305/) (2023)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Score-Based Diffusion Models as Principled Priors for Inverse Imaging](https://mlsanity.com/paper/2304.11751/) (2023)
- [Noisier2Noise: Learning to Denoise from Unpaired Noisy Data](https://mlsanity.com/paper/1910.11908/) (2019)
- [Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks](https://mlsanity.com/paper/1810.03982/) (2018)
- [Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Images](https://mlsanity.com/paper/2106.07009/) (2021)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [Restoration-Degradation Beyond Linear Diffusions: A Non-Asymptotic Analysis For DDIM-Type Samplers](https://mlsanity.com/paper/2303.03384/) (2023)
- [From Denoising to Compressed Sensing](https://mlsanity.com/paper/1406.4175/) (2014)
- [Consistent Diffusion Models: Mitigating Sampling Drift by Learning to be Consistent](https://mlsanity.com/paper/2302.09057/) (2023)
- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [SKM-TEA: A Dataset for Accelerated MRI Reconstruction with Dense Image Labels for Quantitative Clinical Evaluation](https://mlsanity.com/paper/2203.06823/) (2022)
- [Solving Inverse Problems with Score-Based Generative Priors learned from Noisy Data](https://mlsanity.com/paper/2305.01166/) (2023)
- [GSURE-Based Diffusion Model Training with Corrupted Data](https://mlsanity.com/paper/2305.13128/) (2023)
- [A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise](https://mlsanity.com/paper/2205.10278/) (2022)

## Built on by (most cited first, 30 of 32)

- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Learning Diffusion Priors from Observations by Expectation Maximization](https://mlsanity.com/paper/2405.13712/) (2024)
- [Warped Diffusion: Solving Video Inverse Problems with Image Diffusion Models](https://mlsanity.com/paper/2410.16152/) (2024)
- [Ambient Diffusion Omni: Training Good Models with Bad Data](https://mlsanity.com/paper/2506.10038/) (2025)
- [Constrained Diffusion with Trust Sampling](https://mlsanity.com/paper/2411.10932/) (2024)
- [Measurement Score-Based Diffusion Model](https://mlsanity.com/paper/2505.11853/) (2025)
- [Integrating Intermediate Layer Optimization and Projected Gradient Descent for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2505.20789/) (2025)
- [DiffEM: Learning from Corrupted Data with Diffusion Models via Expectation Maximization](https://mlsanity.com/paper/2510.12691/) (2025)
- [Inverse Flow and Consistency Models](https://mlsanity.com/paper/2502.11333/) (2025)
- [Self-Supervised Learning from Noisy and Incomplete Data](https://mlsanity.com/paper/2601.03244/) (2026)
- [Robust multi-coil MRI reconstruction via self-supervised denoising](https://mlsanity.com/paper/2411.12919/) (2024)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [Weak Diffusion Priors Can Still Achieve Strong Inverse-Problem Performance](https://mlsanity.com/paper/2601.22443/) (2026)
- [DiffDenoise: Self-Supervised Medical Image Denoising with Conditional Diffusion Models](https://mlsanity.com/paper/2504.00264/) (2025)
- [Diff-Unfolding: A Model-Based Score Learning Framework for Inverse Problems](https://mlsanity.com/paper/2505.11393/) (2025)
- [UGoDIT: Unsupervised Group Deep Image Prior Via Transferable Weights](https://mlsanity.com/paper/2505.11720/) (2025)
- [Fast MRI for All: Bridging Access Gaps by Training without Raw Data](https://mlsanity.com/paper/2411.13022/) (2024)
- [Learning Single Index Models with Diffusion Priors](https://mlsanity.com/paper/2505.21135/) (2025)
- [ADG: Ambient Diffusion-Guided Dataset Recovery for Corruption-Robust Offline Reinforcement Learning](https://mlsanity.com/paper/2505.23871/) (2025)
- [Ambient Dataloops: Generative Models for Dataset Refinement](https://mlsanity.com/paper/2601.15417/) (2026)
- [UPMRI: Unsupervised Parallel MRI Reconstruction via Projected Conditional Flow Matching](https://mlsanity.com/paper/2512.17493/) (2025)
- [Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data](https://mlsanity.com/paper/2601.18728/) (2026)
- [Optimizing Diffusion Priors in Image Reconstruction from a Single Observation](https://mlsanity.com/paper/2604.21066/) (2026)
- [Measurement Score-Based MRI Reconstruction with Automatic Coil Sensitivity Estimation](https://mlsanity.com/paper/2509.18402/) (2025)
- [Patch-Based Diffusion for Data-Efficient, Radiologist-Preferred MRI Reconstruction](https://mlsanity.com/paper/2509.21531/) (2025)
- [Extendable Generalization Self-Supervised Diffusion for Low-Dose CT Reconstruction](https://mlsanity.com/paper/2509.23885/) (2025)
- [Robust Super-Capacity SRS Channel Inpainting via Diffusion Models](https://mlsanity.com/paper/2510.26097/) (2025)
- [Annealed Langevin Posterior Sampling (ALPS): A Rapid Algorithm for Image Restoration with Multiscale Energy Models](https://mlsanity.com/paper/2601.02594/) (2026)
- [SAGE: Subsurface AI-driven Geostatistical Extraction with proxy posterior](https://mlsanity.com/paper/2604.00307/) (2026)
- [Measurement Geometry and Design for Trustworthy Generative Inverse Problems](https://mlsanity.com/paper/2606.02309/) (2026)

## Abstract (arXiv)

We provide a framework for solving inverse problems with diffusion models learned from linearly corrupted data. Firstly, we extend the Ambient Diffusion framework to enable training directly from measurements corrupted in the Fourier domain. Subsequently, we train diffusion models for MRI with access only to Fourier subsampled multi-coil measurements at acceleration factors R= 2,4,6,8. Secondly, we propose Ambient Diffusion Posterior Sampling (A-DPS), a reconstruction algorithm that leverages generative models pre-trained on one type of corruption (e.g. image inpainting) to perform posterior sampling on measurements from a different forward process (e.g. image blurring). For MRI reconstruction in high acceleration regimes, we observe that A-DPS models trained on subsampled data are better suited to solving inverse problems than models trained on fully sampled data. We also test the efficacy of A-DPS on natural image datasets (CelebA, FFHQ, and AFHQ) and show that A-DPS can sometimes outperform models trained on clean data for several image restoration tasks in both speed and performance.