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

This paper proposes a precise signal recovery method with multilayered non-convex regularization, enhancing sparsity/low-rankness for 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 global optimal solution but also offer more accurate estimation than single-layered ones, thanks to their faithful modeling of structured sparsity and low-rankness in high-dimensional signals. However, these functions are known to yield biased solutions (estimated with smaller amplitude values than the true ones). To address this issue, multilayered non-convex regularization functions have been considered, although they face their own challenges: 1) their closed-form proximity operators are unavailable, and 2) convergence may result in a local optimal solution. In this paper, we resolve the two issues with an approach based on epigraphical relaxation (ER). First, ER decomposes a multilayered non-convex regularization function into the outermost function and epigraph constraints for the inner functions, facilitating the computation of proximity operators. Second, the relaxed regularization functions by ER are integrated into a non-convexly regularized convex optimization model to estimate a global optimal solution with less bias. Numerical experiments demonstrate the bias reduction achieved by the proposed method in image recovery and principal component analysis.