Sparse signal recovery by iterative proximal thresholding

Soft thresholding plays a central role in the various signal processing problems in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties. We then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and establish the strong convergence of a proximal thresholding algorithm to solve it. Numerical applications to signal recovery are demonstrated.