We consider the problem of optimizing the sum of a smooth convex function and
a non-smooth convex function using proximal-gradient methods, where an error is
present in the calculation of the gradient of the smooth term or in the
proximity operator with respect to the non-smooth term.

In this paper we study the kernel multiple ridge regression framework, which
we refer to as multi-task regression, using penalization techniques. The
theoretical analysis of this problem shows that the key element appearing for
an optimal calibration is the covariance matrix of the noise between the
different tasks. We present a new algorithm to estimate this covariance matrix,
based on the concept of minimal penalty, which was previously used in the
single-task regression framework to estimate the variance of the noise.

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.

Modeling data with linear combinations of a few elements from a learned
dictionary has been the focus of much recent research in machine learning,
neuroscience and signal processing. For signals such as natural images that
admit such sparse representations, it is now well established that these models
are well suited to restoration tasks. In this context, learning the dictionary
amounts to solving a large-scale matrix factorization problem, which can be
done efficiently with classical optimization tools.

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 use convex relaxation techniques to produce lower bounds on the optimal
value of subset selection problems and generate good approximate solutions. We
then explicitly bound the quality of these relaxations by studying the
approximation ratio of sparse eigenvalue relaxations. Our results are used to
improve the performance of branch-and-bound algorithms to produce exact
solutions to subset selection problems.

Sparse coding--that is, modelling data vectors as sparse linear combinations
of basis elements--is widely used in machine learning, neuroscience, signal
processing, and statistics. This paper focuses on the large-scale matrix
factorization problem that consists of learning the basis set, adapting it to
specific data. Variations of this problem include dictionary learning in signal
processing, non-negative matrix factorization and sparse principal component
analysis.

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 present an extension of sparse PCA, or sparse dictionary learning, where
the sparsity patterns of all dictionary elements are structured and constrained
to belong to a prespecified set of shapes. This \emph{structured sparse PCA} is
based on a structured regularization recently introduced by [1]. While
classical sparse priors only deal with \textit{cardinality}, the regularization
we use encodes higher-order information about the data. We propose an efficient
and simple optimization procedure to solve this problem.