Conditional Mutual Information Based Diffusion Posterior Sampling for Solving Inverse Problems

Inverse problems are prevalent across various disciplines in science and engineering. In the field of computer vision, tasks such as inpainting, deblurring, and super-resolution are commonly formulated as inverse problems. Recently, diffusion models (DMs) have emerged as a promising approach for addressing noisy linear inverse problems, offering effective solutions without requiring additional task-specific training. Specifically, with the prior provided by DMs, one can sample from the posterior by finding the likelihood. Since the likelihood is intractable, it is often approximated in the literature. However, this approximation compromises the quality of the generated images. To overcome this limitation and improve the effectiveness of DMs in solving inverse problems, we propose an information-theoretic approach. Specifically, we maximize the conditional mutual information I(boldsymbolx0; boldsymboly | boldsymbolxt), where boldsymbolx0 represents the reconstructed signal, boldsymboly is the measurement, and boldsymbolxt is the intermediate signal at stage t. This ensures that the intermediate signals boldsymbolxt are generated in a way that the final reconstructed signal boldsymbolx0 retains as much information as possible about the measurement boldsymboly. We demonstrate that this method can be seamlessly integrated with recent approaches and, once incorporated, enhances their performance both qualitatively and quantitatively.