Solution of a Bivariate ℓ1 Regularized Problem

Ilker Bayram

Abstract: 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 ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm. The derived mapping is useful for accelerating convergence of iterative algorithms that aim to solve ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> regularized problems. We discuss how to use the mapping in practice and demonstrate the improvement in convergence rate with numerical experiments.