This seminar presents two recent works focused on sparse signal recovery and inverse problems. The first part introduces the truncated Huber penalty, a non-convex penalty function designed for robust signal recovery. We explore its application in constrained and unconstrained models, proving theoretical properties of the optimal solutions. An efficient algorithm based on the block coordinate descent method is also discussed, along with applications in signal and image processing. The second part covers a generalized Tikhonov regularization framework with spatially varying weights estimated via a neural network. This end-to-end approach integrates adaptive parameter estimation, improving detail preservation.