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Preface 1 Introduction 2 The Lasso for Linear Models 2.1 Introduction 2.2 The Lasso Estimator 2.3 Cross-Validation and Inference 2.4 Computation of the Lasso Solution 2.4.1 Single Predictor: Soft Thresholding 2.4.2 Multiple Predictors: Cyclic Coordinate Descent 2.4.3 Soft-Thresholding and Orthogonal Bases 2.5 Degrees of Freedom 2.6 Uniqueness of the Lasso Solutions 2.7 A Glimpse at the Theory 2.8 The Nonnegative Garrote 2.9 lq Penalties and Bayes Estimates 2.10 Some Perspective Exercises 3 Generalized Linear Models 3.1 Introduction 3.2 Logistic Regression 3.2.1 Example: Document Classification 3.2.2 Algorithms 3.3 Multiclass Logistic Regression 3.3.1 Example: Handwritten Digits 3.3.2 Algorithms 3.3.3 Grouped-Lasso Multinomial 3.4 Log-Linear Models and the Poisson GLM 3.4.1 Example: Distribution Smoothing 3.5 Cox Proportional Hazards Models 3.5.1 Cross-Validation 3.5.2 Pre-Validation 3.6 Support Vector Machines 3.6.1 Logistic Regression with Separable Data 3.7 Computational Details and glmnet Bibliographic Notes Exercises 4 Generalizations of the Lasso Penalty 4.1 Introduction 4.2 The Elastic Net 4.3 The Group Lasso 4.3.1 Computation for the Group Lasso 4.3.2 Sparse Group Lasso 4.3.3 The Overlap Group Lasso 4.4 Sparse Additive Models and the Group Lasso 4.4.1 Additive Models and Backfitting 4.4.2 Sparse Additive Models and Backfitting 4.4.3 Approaches Using Optimization and the Group Lasso 4.4.4 Multiple Penalization for Sparse Additive Models 4.5 The Fused Lasso 4.5.1 Fitting the Fused Lasso 4.5.1.1 Reparametrization 4.5.1.2 A Path Algorithm 4.5.1.3 A Dual Path Algorithm 4.5.1.4 Dynamic Programming for the Fused Lasso 4.5.2 Trend Filtering 4.5.3 Nearly Isotonic Regression 4.6 Nonconvex Penalties Bibliographic Notes Exercises 5 Optimization Methods 5.1 Introduction 5.2 Convex Optimality Conditions 5.2.1 Optimality for Differentiable Problems 5.2.2 Nondifferentiable Functions and Subgradients 5.3 Gradient Descent 5.3.1 Unconstrained Gradient Descent 5.3.2 Projected Gradient Methods 5.3.3 Proximal Gradient Methods 5.3.4 Accelerated Gradient Methods 5.4 Coordinate Descent 5.4.1 Separability and Coordinate Descent 5.4.2 Linear Regression and the Lasso 5.4.3 Logistic Regression and Generalized Linear Models 5.5 A Simulation Study 5.6 Least Angle Regression 5.7 Alternating Direction Method of Multipliers 5.8 Minorization-Maximization Algorithms 5.9 Biconvexity and Alternating Minimization 5.10 Screening Rules Bibliographic Notes Appendix Exercises 6 Statistical Inference 6.1 The Bayesian Lasso 6.2 The Bootstrap 6.3 Post-Selection Inference for the Lasso 6.3.1 The Covariance Test 6.3.2 A General Scheme for Post-Selection Inference 6.3.2.1 Fixed-Èë Inference for the Lasso 6.3.2.2 The Spacing Test for LAR 6.3.3 What Hypothesis Is Being Tested? 6.3.4 Back to Forward Stepwise Regression 6.4 Inference via a Debiased Lasso 6.5 Other Proposals for Post-Selection Inference Bibliographic Notes Exercises 7 Matrix Decompositions, Approximations, and Completion 7.1 Introduction 7.2 The Singular Value Decomposition 7.3 Missing Data and Matrix Completion 7.3.1 The Netflix Movie Challenge 7.3.2 Matrix Completion Using Nuclear Norm 7.3.3 Theoretical Results for Matrix Completion 7.3.4 Maximum Margin Factorization and Related Methods 7.4 Reduced-Rank Regression 7.5 A General Matrix Regression Framework 7.6 Penalized Matrix Decomposition 7.7 Additive Matrix Decomposition Bibliographic Notes Exercises 8 Sparse Multivariate Methods 8.1 Introduction 8.2 Sparse Principal Components Analysis 8.2.1 Some Background 8.2.2 Sparse Principal Components 8.2.2.1 Sparsity from Maximum Variance 8.2.2.2 Methods Based on Reconstruction 8.2.3 Higher-Rank Solutions 8.2.3.1 Illustrative Application of Sparse PCA 8.2.4 Sparse PCA via Fantope Projection 8.2.5 Sparse Autoencoders and Deep Learning 8.2.6 Some Theory for Sparse PCA 8.3 Sparse Canonical Correlation Analysis 8.3.1 Example: Netflix Movie Rating Data 8.4 Sparse Linear Discriminant Analysis 8.4.1 Normal Theory and Bayes' Rule 8.4.2 Nearest Shrunken Centroids 8.4.3 Fisher's Linear Discriminant Analysis 8.4.3.1 Example: Simulated Data with Five Classes 8.4.4 Optimal Scoring 8.4.4.1 Example: Face Silhouettes 8.5 Sparse Clustering 8.5.1 Some Background on Clustering 8.5.1.1 Example: Simulated Data with Six Classes 8.5.2 Sparse Hierarchical Clustering 8.5.3 Sparse K-Means Clustering 8.5.4 Convex Clustering Bibliographic Notes Exercises 9 Graphs and Model Selection 9.1 Introduction 9.2 Basics of Graphical Models 9.2.1 Factorization and Markov Properties 9.2.1.1 Factorization Property 9.2.1.2 Markov Property 9.2.1.3 Equivalence of Factorization and Markov Properties 9.2.2 Some Examples 9.2.2.1 Discrete Graphical Models 9.2.2.2 Gaussian Graphical Models 9.3 Graph Selection via Penalized Likelihood 9.3.1 Global Likelihoods for Gaussian Models 9.3.2 Graphical Lasso Algorithm 9.3.3 Exploiting Block-Diagonal Structure 9.3.4 Theoretical Guarantees for the Graphical Lasso 9.3.5 Global Likelihood for Discrete Models 9.4 Graph Selection via Conditional Inference 9.4.1 Neighborhood-Based Likelihood for Gaussians 9.4.2 Neighborhood-Based Likelihood for Discrete Models 9.4.3 Pseudo-Likelihood for Mixed Models 9.5 Graphical Models with Hidden Variables Bibliographic Notes Exercises 10 Signal Approximation and Compressed Sensing 10.1 Introduction 10.2 Signals and Sparse Representations 10.2.1 Orthogonal Bases 10.2.2 Approximation in Orthogonal Bases 10.2.3 Reconstruction in Overcomplete Bases 10.3 Random Projection and Approximation 10.3.1 Johnson¨CLindenstrauss Approximation 10.3.2 Compressed Sensing 10.4 Equivalence between lo and l1 Recovery 10.4.1 Restricted Nullspace Property 10.4.2 Sufficient Conditions for Restricted Nullspace 10.4.3 Proofs 10.4.3.1 Proof of Theorem 10.1 10.4.3.2 Proof of Proposition 10.1 Bibliographic Notes Exercises 11 Theoretical Results for the Lasso 11.1 Introduction 11.1.1 Types of Loss Functions 11.1.2 Types of Sparsity Models 11.2 Bounds on Lasso l2-Error 11.2.1 Strong Convexity in the Classical Setting 11.2.2 Restricted Eigenvalues for Regression 11.2.3 A Basic Consistency Result 11.3 Bounds on Prediction Error 11.4 Support Recovery in Linear Regression 11.4.1 Variable-Selection Consistency for the Lasso 11.4.1.1 Some Numerical Studies 11.5 Beyond the Basic Lasso Bibliographic Notes Exercises Bibliography Author Index Index