Inverse inference, or "brain reading", is a recent paradigm for analyzing
functional magnetic resonance imaging (fMRI) data, based on pattern recognition
and statistical learning. By predicting some cognitive variables related to
brain activation maps, this approach aims at decoding brain activity. Inverse
inference takes into account the multivariate information between voxels and is
currently the only way to assess how precisely some cognitive information is
encoded by the activity of neural populations within the whole brain.

Sparse coding consists in representing signals as sparse linear combinations
of atoms selected from a dictionary. We consider an extension of this framework
where the atoms are further assumed to be embedded in a tree. This is achieved
using a recently introduced tree-structured sparse regularization norm, which
has proven useful in several applications. This norm leads to regularized
problems that are difficult to optimize, and we propose in this paper efficient
algorithms for solving them.

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.