Abstract :

Many estimation problems amount to minimizing an objective function composed of a quadratic data-fidelity term and a general regularization term. It is widely accepted that the minimizers obtained using nonsmooth and/or nonconvex regularization terms are frequently good estimates. However, very few facts are known on the ways to control properties of these minimizers. This work is dedicated to the stability of the minimizers of such nonsmooth and/or nonconvex objective functions. It consists of two parts: in this part, we focus on general local minimizers, whereas in a second part, we derive results on global minimizers. Here we demonstrate that the data domain contains an open, dense subset whose elements give rise to local and global minimizers which are necessarily strict. Moreover, we show that the relevant minimizers are stable under variations of the data.