Statistical learning with sparsity: the lasso and generalizations - Boca Raton CRC Press 2015 - xiii, 351 p. - Chapman & Hall/CRC Monographs on Statistics & Applied Probability .

Table of Contents:

1. Introduction

2. The Lasso for Linear Models

Introduction

The Lasso Estimator

Cross-Validation and Inference

Computation of the Lasso Solution

Degrees of Freedom

Uniqueness of the Lasso Solutions

A Glimpse at the Theory

The Nonnegative Garrote

ℓq Penalties and Bayes Estimates

Some Perspective

3. Generalized Linear Models

Introduction

Logistic Regression

Multiclass Logistic Regression

Log-Linear Models and the Poisson GLM

Cox Proportional Hazards Models

Support Vector Machines

Computational Details and glmnet

4. Generalizations of the Lasso Penalty

Introduction

The Elastic Net

The Group Lasso

Sparse Additive Models and the Group Lasso

The Fused Lasso

Nonconvex Penalties

5. Optimization Methods

Introduction

Convex Optimality Conditions

Gradient Descent

Coordinate Descent

A Simulation Study

Least Angle Regression

Alternating Direction Method of Multipliers

Minorization-Maximization Algorithms

Biconvexity and Alternating Minimization

Screening Rules

6. Statistical Inference

The Bayesian Lasso

The Bootstrap

Post-Selection Inference for the Lasso

Inference via a Debiased Lasso

Other Proposals for Post-Selection Inference

7. Matrix Decompositions, Approximations, and Completion

Introduction

The Singular Value Decomposition

Missing Data and Matrix Completion

Reduced-Rank Regression

A General Matrix Regression Framework

Penalized Matrix Decomposition

Additive Matrix Decomposition

8. Sparse Multivariate Methods

Introduction

Sparse Principal Components Analysis

Sparse Canonical Correlation Analysis

Sparse Linear Discriminant Analysis

Sparse Clustering

9. Graphs and Model Selection

Introduction

Basics of Graphical Models

Graph Selection via Penalized Likelihood

Graph Selection via Conditional Inference

Graphical Models with Hidden Variables

10. Signal Approximation and Compressed Sensing

Introduction

Signals and Sparse Representations

Random Projection and Approximation

Equivalence between ℓ0 and ℓ1 Recovery

11. Theoretical Results for the Lasso

Introduction

Bounds on Lasso ℓ2-error

Bounds on Prediction Error

Support Recovery in Linear Regression

Beyond the Basic Lasso

Discover New Methods for Dealing with High-Dimensional Data

A sparse statistical model has only a small number of nonzero parameters or weights; therefore, it is much easier to estimate and interpret than a dense model. Statistical Learning with Sparsity: The Lasso and Generalizations presents methods that exploit sparsity to help recover the underlying signal in a set of data.

Top experts in this rapidly evolving field, the authors describe the lasso for linear regression and a simple coordinate descent algorithm for its computation. They discuss the application of l1 penalties to generalized linear models and support vector machines, cover generalized penalties such as the elastic net and group lasso, and review numerical methods for optimization. They also present statistical inference methods for fitted (lasso) models, including the bootstrap, Bayesian methods, and recently developed approaches. In addition, the book examines matrix decomposition, sparse multivariate analysis, graphical models, and compressed sensing. It concludes with a survey of theoretical results for the lasso.

In this age of big data, the number of features measured on a person or object can be large and might be larger than the number of observations. This book shows how the sparsity assumption allows us to tackle these problems and extract useful and reproducible patterns from big datasets. Data analysts, computer scientists, and theorists will appreciate this thorough and up-to-date treatment of sparse statistical modeling.

(https://www.crcpress.com/Statistical-Learning-with-Sparsity-The-Lasso-and-Generalizations/Hastie-Tibshirani-Wainwright/p/book/9781498712163)

9781498712163

Mathematical statistics

Least squares

Linear models (Statistics)

Proof theory

519.5 / H2S8