  Sparse estimation methods are aimed at using or obtaining parsimonious
representations of data or models. While naturally cast as a combinatorial
optimization problem, variable or feature selection admits a convex relaxation
through the regularization by the $\ell_1$-norm. In this paper, we consider
situations where we are not only interested in sparsity, but where some
structural prior knowledge is available as well. We show that the $\ell_1$-norm
can then be extended to structured norms built on either disjoint or
overlapping groups of variables, leading to a flexible framework that can deal
with various structures. We present applications to unsupervised learning, for
structured sparse principal component analysis and hierarchical dictionary
learning, and to supervised learning in the context of non-linear variable
selection.
