Deep Block Proximal Linearised Minimisation Algorithm for Non-convex Inverse Problems

Image restoration is typically addressed through non-convex inverse problems, which are often solved using first-order block-wise splitting methods. In this paper, we consider a general type of non-convex optimisation model that captures many inverse image problems and present an inertial block proximal linearised minimisation (iBPLM) algorithm. Our new method unifies the Jacobi-type parallel and the Gauss-Seidel-type alternating update rules, and extends beyond these approaches. The inertial technique is also incorporated into each block-wise subproblem update, which can accelerate numerical convergence. Furthermore, we extend this framework with a plug-and-play variant (PnP-iBPLM) that integrates deep gradient denoisers, offering a flexible and robust solution for complex imaging tasks. We provide comprehensive theoretical analysis, demonstrating both subsequential and global convergence of the proposed algorithms. To validate our methods, we apply them to multi-block dictionary learning problems in image denoising and deblurring. Experimental results show that both iBPLM and PnP-iBPLM significantly enhance numerical performance and robustness in these applications.