Distribution Shift Estimation in Imaging Inverse Problems

Diffusion models have emerged as a powerful image priors for inverse problems. However, their performance can degrade significantly when there is a distribution shift between training and testing datasets. Quantifying distribution shifts is thus essential for ensuring the reliability of diffusion model priors. Existing shift detection methods typically require access to clean data, which is often unavailable in inverse problems. We propose a measurement-domain KL divergence estimator for linear inverse problems, enabling shift quantification directly from corrupted measurements. We prove that the proposed metric can recover the KL divergence between training and test distributions using only noisy observations and pretrained diffusion models. Our method provides a practical, unsupervised approach for detecting distribution shifts in inverse problems such as image deblurring—without relying on clean images.