Sparsity-Enhanced Multilayered Non-Convex Regularization with Epigraphical Relaxation for Debiased Signal Recovery

This paper proposes a precise signal recovery method utilizing a composition of multiple non-convex regularization functions, termed multilayered non-convex regularization functions, to enhance sparsity and low-rankness in high-dimensional signals, including images and videos. In optimization-based signal recovery, multilayered convex regularization functions based on the l1 and nuclear norms not only guarantee a globally optimal solution but also improve estimation accuracy over single-layered ones by effectively modeling structured sparsity and low-rankness across various domains, such as spatial, spectral, and temporal domains. However, they often yield biased solutions with underestimated amplitudes. To address this underestimation, multilayered non-convex regularization functions have been explored but face challenges: 1) their closed-form proximity operators are often unavailable, and 2) convergence may lead to local optima. To overcome these issues, we propose the ER-LiGME model, which integrates epigraphical relaxation (ER) with the linearly-involved generalized Moreau-enhanced (LiGME) model. ER decomposes a multilayered composite function into its outermost function and epigraph constraints, facilitating proximity operator computation. Meanwhile, the LiGME model transforms convex regularization functions into non-convex ones while preserving convexity of the optimization model. By leveraging ER to reformulate multilayered convex regularization functions and integrating the LiGME model, the proposed approach provides a framework for diverse multilayered non-convex regularization functions that ensure global optimality, reducing bias and high-accuracy of high-dimensional signal recovery. Numerical experiments demonstrate the proposed method’s effectiveness in image recovery and principal component analysis.