Solution of a Bivariate Regularized Problem

We derive the mapping that takes an observation vector to the minimizer of a bivariate cost consisting of the sum of a quadratic data fidelity term and an [Formula Omitted] norm. The derived mapping is useful for accelerating convergence of iterative algorithms that aim to solve [Formula Omitted] regularized problems. We discuss how to use the mapping in practice and demonstrate the improvement in convergence rate with numerical experiments.