A Unified View of Exact Continuous Penalties for $\ell_2$-$\ell_0$ Minimization

International audienceNumerous nonconvex continuous penalties have been proposed to approach the l0 pseudo-norm for optimization purpose. Apart from the theoretical results for convex l1 relaxation under restrictive hypothesis, only few works have been devoted to analyze the consistency, in terms of minimizers, between the l0-regularized least square functional and relaxed ones using continuous approximations. In this context, two questions are of fundamental importance: does relaxed functionals preserve global minimizers of the initial one? Does this approximation introduce unwanted new (local) minimizers? In this paper we answer these questions by deriving necessary and sufficient conditions on such l0 continuous approximations in order that each minimizer of the underlying relaxation is also a minimizer of the l2-l0 functional and that all the global minimizers of the initial functional are preserved. Hence, a general class of penalties is provided giving a unified view of exact continuous approximations of the l0-norm within the l2-l0 minimization framework. As the inferior limit of this class of penalties, we get the recently proposed CEL0 penalty. Finally, state of the art penalties, such as MCP, SCAD or Capped-l1, are analyzed according to the proposed class of exact continuous penalties