  This paper provides a set of sensitivity analysis and activity identification
results for a class of convex functions with a strong geometric structure, that
we coined "mirror-stratifiable". These functions are such that there is a
bijection between a primal and a dual stratification of the space into
partitioning sets, called strata. This pairing is crucial to track the strata
that are identifiable by solutions of parametrized optimization problems or by
iterates of optimization algorithms. This class of functions encompasses all
regularizers routinely used in signal and image processing, machine learning,
and statistics. We show that this "mirror-stratifiable" structure enjoys a nice
sensitivity theory, allowing us to study stability of solutions of optimization
problems to small perturbations, as well as activity identification of
first-order proximal splitting-type algorithms. Existing results in the
literature typically assume that, under a non-degeneracy condition, the active
set associated to a minimizer is stable to small perturbations and is
identified in finite time by optimization schemes. In contrast, our results do
not require any non-degeneracy assumption: in consequence, the optimal active
set is not necessarily stable anymore, but we are able to track precisely the
set of identifiable strata.We show that these results have crucial implications
when solving challenging ill-posed inverse problems via regularization, a
typical scenario where the non-degeneracy condition is not fulfilled. Our
theoretical results, illustrated by numerical simulations, allow to
characterize the instability behaviour of the regularized solutions, by
locating the set of all low-dimensional strata that can be potentially
identified by these solutions.
