Estimation and testing under sparsity : ecole d'ete de probabilites de Saint-Flour xlv - 2015 / Sara Van de Geer.

Includes bibliographical references and index.

1. Introduction --
2. The Lasso --
3. The square-root Lasso --
4. The bias of the Lasso and worst possible sub-directions --
5. Confidence intervals using the Lasso --
6. Structured sparsity --
7. General loss with norm-penalty --
8. Empirical process theory for dual norms --
9. Probability inequalities for matrices --
10. Inequalities for the centred empirical risk and its derivative --
11. The margin condition --
12. Some worked-out examples --
13. Brouwer's fixed point theorem and sparsity --
14. Asymptotically linear estimators of the precision matrix --
15. Lower bounds for sparse quadratic forms --
16. Symmetrization, contraction and concentration --
17. Chaining including concentration --
18. Metric structure of convex hulls.

Taking the Lasso method as its starting point, this book describes the main ingredients needed to study general loss functions and sparsity-inducing regularizers. It also provides a semi-parametric approach to establishing confidence intervals and tests. Sparsity-inducing methods have proven to be very useful in the analysis of high-dimensional data. Examples include the Lasso and group Lasso methods, and the least squares method with other norm-penalties, such as the nuclear norm. The illustrations provided include generalized linear models, density estimation, matrix completion and sparse principal components. Each chapter ends with a problem section. The book can be used as a textbook for a graduate or PhD course.