Figure 1. Illustration of the Measurement Score-based diffusion Model (MSM) for training and sampling using subsampled data. Training: MSM is trained solely on degraded measurements. Diffusion noise is added to these measurements, and the model learns to denoise them. Sampling: At each diffusion step, MSM randomly subsamples the current full-measurement iterate, denoises the resulting partial measurement, and aggregates multiple outputs. A weighting vector compensates for overlapping contributions across partial measurements.
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 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.
Measurement Score-Based Sampling Algorithm
Algorithms 1. At each step, we sample \( w \) random subsampling masks \( \boldsymbol{S}^{(i)} \) to form partial measurements. Each partial measurement is denoised using the partial measurement score \( \mathsf{S}_{\theta} \). The denoised outputs are reprojected to the full measurement space via \( \boldsymbol{S}^{(i)\top} \) and combined using a reweighting vector \( \boldsymbol{W} \). This process iteratively updates the full measurement iterate \( \mathbf{z}_t \) toward the final fully-sampled measurement \( \mathbf{z}_0 \).
Figure 2. Illustration of the Measurement Score-based diffusion Model (MSM) for training and sampling using subsampled MRI measurements. Here, \( \boldsymbol{C} \) denotes the coil sensitivity operator and \( \boldsymbol{F} \) denotes the Fourier transform.
Numerical Evaluations
Figure 3. Generated samples from MSM trained under three degradation settings (first row: training data; second row: samples generated by models trained on the corresponding data). Note how despite never seeing ground-truth data, MSM can generated high-quality images.
Figure 4. Visual comparison of methods trained on subsampled data for inverse problems.
Figure 5. Visual comparison between MSM trained on subsampled data and diffusion-based inverse problem solvers trained on clean data.
