# Measurement Score-Based Diffusion Model

arXiv: https://arxiv.org/abs/2505.11853 · HTML: https://mlsanity.com/paper/2505.11853/

## Key facts

- Cited by 8 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 31 of its 45 references that are in the map.
- First posted on arXiv 17 May 2025.

## 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 31)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model](https://mlsanity.com/paper/2212.00490/) (2022)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Ambient Diffusion: Learning Clean Distributions from Corrupted Data](https://mlsanity.com/paper/2305.19256/) (2023)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming](https://mlsanity.com/paper/1310.3787/) (2013)
- [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)
- [DDM2: Self-Supervised Diffusion MRI Denoising with Generative Diffusion Models](https://mlsanity.com/paper/2302.03018/) (2023)
- [Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data](https://mlsanity.com/paper/2404.10177/) (2024)
- [Accelerating Diffusion Models with Parallel Sampling: Inference at Sub-Linear Time Complexity](https://mlsanity.com/paper/2405.15986/) (2024)
- [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)
- [Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data](https://mlsanity.com/paper/2403.08728/) (2024)
- [A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise](https://mlsanity.com/paper/2205.10278/) (2022)
- [Online Deep Equilibrium Learning for Regularization by Denoising](https://mlsanity.com/paper/2205.13051/) (2022)
- [An Expectation-Maximization Algorithm for Training Clean Diffusion Models from Corrupted Observations](https://mlsanity.com/paper/2407.01014/) (2024)
- [Conditioning non-linear and infinite-dimensional diffusion processes](https://mlsanity.com/paper/2402.01434/) (2024)
- [Clean self-supervised MRI reconstruction from noisy, sub-sampled training data with Robust SSDU](https://mlsanity.com/paper/2210.01696/) (2022)
- [SPICER: Self-Supervised Learning for MRI with Automatic Coil Sensitivity Estimation and Reconstruction](https://mlsanity.com/paper/2210.02584/) (2022)
- [A Regularized Conditional GAN for Posterior Sampling in Image Recovery Problems](https://mlsanity.com/paper/2210.13389/) (2022)
- [Random Walks with Tweedie: A Unified View of Score-Based Diffusion Models](https://mlsanity.com/paper/2411.18702/) (2024)

## Built on by (most cited first, 8 of 8)

- [Ambient Diffusion Policy: Imitation Learning from Suboptimal Data in Robotics](https://mlsanity.com/paper/2606.12365/) (2026)
- [Normalization-equivariant Diffusion Models: Learning Posterior Samplers From Noisy And Partial Measurements](https://mlsanity.com/paper/2510.11964/) (2025)
- [Measurement Score-Based MRI Reconstruction with Automatic Coil Sensitivity Estimation](https://mlsanity.com/paper/2509.18402/) (2025)
- [Observation-Aligned Mask Priors for Learning Physical Fields from Authentic Occlusions](https://mlsanity.com/paper/2605.16818/) (2026)
- [Trajectory Constraints for Imaging Inverse Problems](https://mlsanity.com/paper/2605.29012/) (2026)
- [CLIMB-flow: Coupled Linear Inverse posterior sampling via Multiscale-Based flow](https://mlsanity.com/paper/2609.33834/) (2026)
- [Learning Field Reconstruction from Incomplete Data by Globally Correcting Local Estimates](https://mlsanity.com/paper/2610.05375/) (2026)
- [Ambient Discrete Diffusion: Using the Wrong Data at the Right Time for Data Efficient Learning](https://mlsanity.com/paper/2610.12340/) (2026)

## Abstract (arXiv)

Diffusion models are widely used in applications ranging from image generation to inverse problems. However, training diffusion models typically requires clean ground-truth images, which are unavailable in many applications. We introduce the Measurement Score-based diffusion Model (MSM), a novel framework that learns partial measurement scores using only noisy and subsampled measurements. MSM models the distribution of full measurements as an expectation over partial scores induced by randomized subsampling. To make the MSM representation computationally efficient, we also develop a stochastic sampling algorithm that generates full images by using a randomly selected subset of partial scores at each step. We additionally propose a new posterior sampling method for solving inverse problems that reconstructs images using these partial scores. We provide a theoretical analysis that bounds the Kullback-Leibler divergence between the distributions induced by full and stochastic sampling, establishing the accuracy of the proposed algorithm. We demonstrate the effectiveness of MSM on natural images and multi-coil MRI, showing that it can generate high-quality images and solve inverse problems -- all without access to clean training data. Code is available at https://github.com/wustl-cig/MSM.