Measurement Score-Based Diffusion Model
Chicago Y. Park, Shirin Shoushtari, Hongyu An, Ulugbek S. Kamilov
TL;DR
This paper introduces the Measurement Score-based diffusion Model (MSM), a new framework that learns from noisy, subsampled measurements to generate high-quality images and solve inverse problems without needing clean training data.
Contribution
MSM is the first diffusion model that learns from partial measurement scores using only noisy and subsampled data, enabling image generation and inverse problem solving without clean ground-truth images.
Findings
MSM effectively generates high-quality images from noisy measurements.
MSM successfully solves inverse problems like MRI reconstruction.
Theoretical bounds validate the accuracy of the stochastic sampling algorithm.
Abstract
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…
Peer Reviews
Decision·ICLR 2026 Poster
Addresses a clinically important setting in MRI, where fully sampled data are rarely available, and demonstrates strong performance for both natural image inpainting and MRI reconstruction. Provides a theoretical formulation which links subsampled measurement scores to the full measurement score, thereby the authors extend prior score-decomposition ideas (patch-based) to the measurement setting.
The method is developed and demonstrated primarily for subsampled measurements (e.g., incomplete k-space or inpainting), where spatial subsampling applies naturally. It is less clear how the formulation extends to other partial-measurement settings such as limited-angle tomography in CT, where missing data correspond to absent projection views rather than spatially masked measurements. Thus, the main practical advantage currently appears in MRI-like scenarios. The paper lacks self-consistency i
The paper attempts to solve an important problem in generative inverse problem solvers that has been previously explored by earlier work. The paper provides an original method for learning measurement scores. The method is communicated clearly and is backed by solid experiments in which they compare their technique to existing SOTA self-supervised generative and end-to-end techniques. Their experiments are convincing that their technique is better than existing self-supervised approaches.
There are some experiments that I think would help strengthen the paper. An ablation over measurement noise would be good to show how performance varies over more than just a single noise level. This goes for both training and inference time. At a bare minimum we should see performance of the posterior solver at the same noise level as the training measurement noise (apologies if I misread and this is the case). Along this same idea, It would also be good to make sure that when you are running
The proposed MSM framework addresses the same problem as other works, such as ambient diffusion, G-SURE, etc, but is quite novel in terms of the methodology and with broader applicability. MSM can also be trained with noiseless subsampled measurements, offering a useful solution in practice, where the measurements are often noisy. Theorem 1 adds more validity to the proposed stochastic approximation of MSM, which shows that the approximation becomes more accurate with more Monte-Carlo samples,
The paper presents a very practical and impactful methodology overall, but the method needs many clarifications regarding the MSM score and why it is defined as such, which I believe lies at the core of the proposed methodology, its novelty, and empirical effectiveness. Also, additional experiments are required to verify the arguments, including a fairer comparison in the case of inverse problem solving. Also, there are a lot of instances in the main text where the explanations are very poor and
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Taxonomy
TopicsMedical Image Segmentation Techniques · Advanced Neuroimaging Techniques and Applications · MRI in cancer diagnosis
MethodsDiffusion