Estimation and Testing Under Sparsity: École d'Été de Probabilités de Saint-Flour XLV – 2015

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Springer, Jun 28, 2016 - Mathematics - 274 pages
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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Contents

1 Introduction
1
2 The Lasso
5
3 The SquareRoot Lasso
26
4 The Bias of the Lasso and Worst Possible Subdirections
41
5 Confidence Intervals Using the Lasso
61
6 Structured Sparsity
75
7 General Loss with NormPenalty
103
8 Empirical Process Theory for Dual Norms
121
12 Some WorkedOut Examples
166
13 Brouwers Fixed Point Theorem and Sparsity
199
14 Asymptotically Linear Estimators of the Precision Matrix
215
15 Lower Bounds for Sparse Quadratic Forms
222
16 Symmetrization Contraction and Concentration
233
17 Chaining Including Concentration
239
18 Metric Structure of Convex Hulls
254
References
267

9 Probability Inequalities for Matrices
133
10 Inequalities for the Centred Empirical Risk and Its Derivative
139
11 The Margin Condition
151
Author Index
271
Index
273
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