  In this paper, we are concerned with regularized regression problems where
the prior regularizer is a proper lower semicontinuous and convex function
which is also partly smooth relative to a Riemannian submanifold. This
encompasses as special cases several known penalties such as the Lasso
($\ell^1$-norm), the group Lasso ($\ell^1-\ell^2$-norm), the
$\ell^\infty$-norm, and the nuclear norm. This also includes so-called
analysis-type priors, i.e. composition of the previously mentioned penalties
with linear operators, typical examples being the total variation or fused
Lasso penalties.We study the sensitivity of any regularized minimizer to
perturbations of the observations and provide its precise local
parameterization.Our main sensitivity analysis result shows that the predictor
moves locally stably along the same active submanifold as the observations
undergo small perturbations. This local stability is a consequence of the
smoothness of the regularizer when restricted to the active submanifold, which
in turn plays a pivotal role to get a closed form expression for the variations
of the predictor w.r.t. observations. We also show that, for a variety of
regularizers, including polyhedral ones or the group Lasso and its analysis
counterpart, this divergence formula holds Lebesgue almost everywhere.When the
perturbation is random (with an appropriate continuous distribution), this
allows us to derive an unbiased estimator of the degrees of freedom and of the
risk of the estimator prediction.Our results hold true without requiring the
design matrix to be full column rank.They generalize those already known in the
literature such as the Lasso problem, the general Lasso problem (analysis
$\ell^1$-penalty), or the group Lasso where existing results for the latter
assume that the design is full column rank.
