Convex implicit multi-scale (i-MuSE) energy framework: bridging compressed sensing and diffusion models

Motivation: Deep reconstruction algorithms do not have theoretical guarantees enjoyed by compressive sensing. Moreover, memory demands of unrolled methods and contraction constraints in Plug-and-Play (PnP) methods restrict their application to large-scale problems. Goal(s): To develop a fast PnP algorithm with guaranteed convergence to global minima and performance comparable to unrolled methods. Approach: We introduce i-MuSE, an implicit multi-scale energy that smoothly approximates the negative log-prior. A local monotone constraint is added to the gradient to ensure local convexity of the energy. Results: i-MuSE outperforms PnP methods and matches unrolled methods. It also offers improved generalizability. Convexity constraint improves convergence and performance at high accelerations. Impact: i-MuSE offers a memory-efficient alternative to unrolled models while guaranteeing convergence, offering better generalization performance, and facilitating fast optimization algorithms. It can also perform posterior sampling, like diffusion models, to estimate uncertainty.