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Estimation and Testing Under Sparsity

École d'Été de Probabilités de Saint-Flour XLV – 2015

  • Book
  • © 2016

Overview

Authors:
  1. Sara van de Geer

    1. Seminar für Statistik HGG 24.1, ETH Zentrum, Zürich, Switzerland

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  • 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
  • A theoretical basis for sparsity-inducing methods is provided, together with ways to build confidence intervals and tests
  • The focus is on common features of methods for high-dimensional data and, as such, a potential starting point is given for the analysis of other methods not treated in the book

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2159)

Part of the book sub series: École d'Été de Probabilités de Saint-Flour (LNMECOLE)

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Table of contents (18 chapters)

  1. Front Matter

    Pages i-xiii
    Download chapter PDF

  2. Introduction

    • Sara van de Geer
    Pages 1-4

  3. The Lasso

    • Sara van de Geer
    Pages 5-25

  4. The Square-Root Lasso

    • Sara van de Geer
    Pages 27-39

  5. The Bias of the Lasso and Worst Possible Sub-directions

    • Sara van de Geer
    Pages 41-60

  6. Confidence Intervals Using the Lasso

    • Sara van de Geer
    Pages 61-74

  7. Structured Sparsity

    • Sara van de Geer
    Pages 75-101

  8. General Loss with Norm-Penalty

    • Sara van de Geer
    Pages 103-119

  9. Empirical Process Theory for Dual Norms

    • Sara van de Geer
    Pages 121-131

  10. Probability Inequalities for Matrices

    • Sara van de Geer
    Pages 133-137

  11. Inequalities for the Centred Empirical Risk and Its Derivative

    • Sara van de Geer
    Pages 139-150

  12. The Margin Condition

    • Sara van de Geer
    Pages 151-165

  13. Some Worked-Out Examples

    • Sara van de Geer
    Pages 167-197

  14. Brouwer’s Fixed Point Theorem and Sparsity

    • Sara van de Geer
    Pages 199-214

  15. Asymptotically Linear Estimators of the Precision Matrix

    • Sara van de Geer
    Pages 215-221

  16. Lower Bounds for Sparse Quadratic Forms

    • Sara van de Geer
    Pages 223-231

  17. Symmetrization, Contraction and Concentration

    • Sara van de Geer
    Pages 233-238

  18. Chaining Including Concentration

    • Sara van de Geer
    Pages 239-253

  19. Metric Structure of Convex Hulls

    • Sara van de Geer
    Pages 255-266

  20. Back Matter

    Pages 267-276
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Keywords

About this book

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.

Reviews

“This book is presented as a series of lecture notes on the theory of penalized estimators under sparsity. … The level of detail is high, and almost all proofs are given in full, with discussion. Each chapter ends with a section of problems, which could be used in a study setting to improve understanding of the proofs.” (Andrew Duncan A. C. Smith, Mathematical Reviews, August, 2017)


“The book provides several examples and illustrations of the methods presented and discussed, while each of its 17 chapters ends with a problem section. Thus, it can be used as textbook for students mainly at postgraduate level.” (Christina Diakaki, zbMATH 1362.62006, 2017)

Authors and Affiliations

  • Seminar für Statistik HGG 24.1, ETH Zentrum, Zürich, Switzerland

    Sara van de Geer

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