  In this paper, we investigate in a unified way the structural properties of
solutions to inverse problems. These solutions are regularized by the generic
class of semi-norms defined as a decomposable norm composed with a linear
operator, the so-called analysis type decomposable prior. This encompasses
several well-known analysis-type regularizations such as the discrete total
variation (in any dimension), analysis group-Lasso or the nuclear norm. Our
main results establish sufficient conditions under which uniqueness and
stability to a bounded noise of the regularized solution are guaranteed. Along
the way, we also provide a strong sufficient uniqueness result that is of
independent interest and goes beyond the case of decomposable norms.
