  We consider the empirical risk minimization problem for linear supervised
learning, with regularization by structured sparsity-inducing norms. These are
defined as sums of Euclidean norms on certain subsets of variables, extending
the usual $\ell_1$-norm and the group $\ell_1$-norm by allowing the subsets to
overlap. This leads to a specific set of allowed nonzero patterns for the
solutions of such problems. We first explore the relationship between the
groups defining the norm and the resulting nonzero patterns, providing both
forward and backward algorithms to go back and forth from groups to patterns.
This allows the design of norms adapted to specific prior knowledge expressed
in terms of nonzero patterns. We also present an efficient active set
algorithm, and analyze the consistency of variable selection for least-squares
linear regression in low and high-dimensional settings.
