  Recent work in signal processing and statistics have focused on defining new
regularization functions, which not only induce sparsity of the solution, but
also take into account the structure of the problem. We present in this paper a
class of convex penalties introduced in the machine learning community, which
take the form of a sum of l_2 and l_infinity-norms over groups of variables.
They extend the classical group-sparsity regularization in the sense that the
groups possibly overlap, allowing more flexibility in the group design. We
review efficient optimization methods to deal with the corresponding inverse
problems, and their application to the problem of learning dictionaries of
natural image patches: On the one hand, dictionary learning has indeed proven
effective for various signal processing tasks. On the other hand, structured
sparsity provides a natural framework for modeling dependencies between
dictionary elements. We thus consider a structured sparse regularization to
learn dictionaries embedded in a particular structure, for instance a tree or a
two-dimensional grid. In the latter case, the results we obtain are similar to
the dictionaries produced by topographic independent component analysis.
