SENSITIVITY ANALYSIS FOR MIRROR-STRATIFIABLE CONVEX FUNCTIONS

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 have called "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 for tracking 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 with respect 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 nondegeneracy condition, the active set associated with a minimizer is stable with respect to small perturbations and is identified in finite time by optimization schemes. In contrast, our results do not require a nondegeneracy 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 in which the nondegeneracy condition is not fulfilled. Our theoretical results, illustrated by numerical simulations, allow us 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.