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Estimation and Testing Under Sparsity
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École d'Été de Probabilités de Saint-Flour XLV ¿ 2015
Paperback
Print on Demand | Lieferzeit: Print on Demand - Lieferbar innerhalb von 3-5 Werktagen I
Artikel-Nr:
9783319327730
Veröffentl:
2016
Einband:
Paperback
Erscheinungsdatum:
29.06.2016
Seiten:
292
Autor:
Sara van de Geer
Gewicht:
446 g
Format:
235x155x16 mm
Serie:
2159, École d'Été de Probabilités de Saint-Flour
Sprache:
Englisch
Beschreibung:
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.
Starting with the popular Lasso method as its prime example, the book then extends to a broad family of estimation methods for high-dimensional data
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.