Sparse multiple kernel learning: Support identification via mirror stratifiability

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. Our analysis rely on the notions of stratification and mirror stratifiability.