Convergence analysis of iteratively reweighted ℓ1$$ {\ell}_1 $$ algorithms for computing the proximal operator of ℓp$$ {\ell}_p $$‐norm

Abstract

The ‐norm with is a widely used nonconvex penalty in compressed sensing, matrix completion, image processing, and among others. Explicit expressions for the proximal operator of ‐norm are only available when and . To handle this, its proximal operator with an arbitrary is frequently evaluated via an iteratively reweighted algorithm (IRL1), which iteratively substitutes ‐norm with its first‐order approximation. In this study, we fully characterize that the IRL1 solution disagrees with the true proximal operator of the ‐norm in certain regions in terms of , an initial value, and the regularization parameter of the proximal operator. Furthermore, an adaptive initial value can be set to ensure that the IRL1 solution always belongs to the proximal operator.