The paper "Proximal Splitting Methods in Signal Processing" by Patrick L. Combettes and Jean-Christophe Pesquet reviews the fundamental properties of proximity operators and presents optimization methods based on these operators. Proximity operators, which are extensions of projection operators onto convex sets, play a crucial role in solving convex optimization problems, particularly in signal processing. The authors discuss the forward-backward algorithm and the Douglas-Rachford algorithm, which are central to proximal splitting methods. These algorithms are shown to unify several well-known algorithms such as iterative thresholding, projected Landweber, and alternating projections. The paper also explores applications of proximal methods in signal recovery and synthesis, including examples like total variation denoising, image deblurring, and compressed sensing. Additionally, the authors address composite problems involving linear transformations and parallel variants of the algorithms for problems with multiple functions. The paper concludes by highlighting the advantages of these algorithms, particularly their ability to handle nondifferentiable objectives commonly encountered in sparse approximation and hard-constrained problems.The paper "Proximal Splitting Methods in Signal Processing" by Patrick L. Combettes and Jean-Christophe Pesquet reviews the fundamental properties of proximity operators and presents optimization methods based on these operators. Proximity operators, which are extensions of projection operators onto convex sets, play a crucial role in solving convex optimization problems, particularly in signal processing. The authors discuss the forward-backward algorithm and the Douglas-Rachford algorithm, which are central to proximal splitting methods. These algorithms are shown to unify several well-known algorithms such as iterative thresholding, projected Landweber, and alternating projections. The paper also explores applications of proximal methods in signal recovery and synthesis, including examples like total variation denoising, image deblurring, and compressed sensing. Additionally, the authors address composite problems involving linear transformations and parallel variants of the algorithms for problems with multiple functions. The paper concludes by highlighting the advantages of these algorithms, particularly their ability to handle nondifferentiable objectives commonly encountered in sparse approximation and hard-constrained problems.