  In statistical machine learning, kernel methods allow to consider infinite
dimensional feature spaces with a computational cost that only depends on the
number of observations. This is usually done by solving an optimization problem
depending on a data fit term and a suitable regularizer. In this paper we
consider feature maps which are the concatenation of a fixed, possibly large,
set of simpler feature maps. The penalty is a sparsity inducing one, promoting
solutions depending only on a small subset of the features. The group lasso
problem is a special case of this more general setting. We show that one of the
most popular optimization algorithms to solve the regularized objective
function, the forward-backward splitting method, allows to perform feature
selection in a stable manner. In particular, we prove that the set of relevant
features is identified by the algorithm after a finite number of iterations if
a suitable qualification condition holds. The main tools used in the proofs are
the notions of stratification and mirror stratifiability.
