Estimation and Testing Under Sparsity [electronic resource] : École d'Été de Probabilités de Saint-Flour XLV – 2015 / by Sara van de Geer
- Lecture Notes in Mathematics, 0075-8434 ; 2159
- 1 Introduction -- 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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