Stability of Minimizers of Regularized Least Squares Objective Functions Ii: Study of the Global Behavior

We address estimation problems where the sought-after solution is defined as the minimizer of an objective function composed of a quadratic data-fidelity term and a regularization term. We especially focus on nonsmooth and/or nonconvex regularization terms because of their ability to yield good estimates. This work is dedicated to the stability of the minimizers of such nonsmooth and/or nonconvex objective functions. It is composed of two parts. In the previous part of this work, we considered general local minimizers. In this part, we derive results on global minimizers. We show that the data domain contains an open, dense subset such that for every data point therein, the objective function has a finite number of local minimizers, and a unique global minimizer which is stable under variations of the data.