Structured sparsity-inducing norms : statistical and algorithmic properties with applications to neuroimaging.

Summary Many areas of industry and applied sciences have witnessed a digital revolution. This has been accompanied by a growth in the volume of data, the processing of which has become a technical challenge. In this context, parsimony has emerged as a central concept in statistical learning. It is indeed natural to want to exploit the available data via a reduced number of parameters. This thesis focuses on a particular and more recent form of parsimony, called structured parsimony. As its name indicates, we will consider situations where, beyond the parsimony alone, we will also have at our disposal a priori knowledge about structural properties of the problem. The objective of this thesis is to analyze the concept of structured parsimony, based on statistical, algorithmic and applied considerations. We start by introducing a family of structured parsimony norms whose statistical aspects are studied in detail. We will then consider dictionary learning, where we will exploit the previously introduced norms in a matrix factorization framework. Different efficient algorithmic tools, such as proximal methods, will then be proposed. Using these tools, we will illustrate on many applications why structured parsimony can be beneficial. These examples include restoration tasks in image processing, hierarchical modeling of textual documents, or prediction of object size from functional magnetic resonance imaging signals.