Measurement Score-Based Diffusion Model

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.