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

By: Material type: TextTextSeries: Lecture notes in mathematics ; 2159. | Ecole d'ete de probabilités de Saint-FlourPublication details: Switzerland : Springer, 2016.Description: xiii, 274 pages ; 24 cmISBN:
  • 9783319327730
Subject(s): DDC classification:
  • 000SA.09 23 V225
Contents:
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.
Summary: 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.
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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.

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