Contributions to sparse methods for complex data analysis

This document is organized around three chapters.that summarize my research activity since 2008, that is, after my PhD thesis. The first chapter provides motivations for my research work. I first depict informally the kind of data statisticians have to deal with in recent application problems. I build on the example of genomics, with which I am familiar, in order to extract the most striking characteristics of modern data that strongly jeopardize the common way of doing statistics. I exhibit important statistical challenges associated with such data and motivate the use of particular tools at the heart of my research preoccupations, which are at the edge of statistics, optimization and machine learning. I then briefly present the main themes of my research and set them in the landscape of the statistical learning community. The second chapter proposes an overview of my contributions to Gaussian Graphical Models (GGM), in terms of modeling and more importantly in terms of inference of the conditional structure associated with such models. After a quick outline of the existing literature and an introduction to the most popular methods in the community,I present my contribution to this field. These contributions were motivated by the need to account for some special features characterizing genomics data and biological networks. The third chapter is dedicated to my works on structured sparse methods. After a brief overview of the basic computational and statistical tools related to sparse regularization, I present four of my contributions to this field. I wish to demonstrate the diversity of these contributions, some being related to algorithmic and computational considerations, while some others are focused on the statistical properties of the methods. All emerged from motivations anchored in applications. A comprehensive curriculum vitae is finally given in the appendix.