  Sparse modeling is a powerful framework for data analysis and processing.
Traditionally, encoding in this framework is done by solving an l_1-regularized
linear regression problem, usually called Lasso. In this work we first combine
the sparsity-inducing property of the Lasso model, at the individual feature
level, with the block-sparsity property of the group Lasso model, where sparse
groups of features are jointly encoded, obtaining a sparsity pattern
hierarchically structured. This results in the hierarchical Lasso, which shows
important practical modeling advantages. We then extend this approach to the
collaborative case, where a set of simultaneously coded signals share the same
sparsity pattern at the higher (group) level but not necessarily at the lower
one. Signals then share the same active groups, or classes, but not necessarily
the same active set. This is very well suited for applications such as source
separation. An efficient optimization procedure, which guarantees convergence
to the global optimum, is developed for these new models. The underlying
presentation of the new framework and optimization approach is complemented
with experimental examples and preliminary theoretical results.
