Statistical learning with sparsity: the lasso and generalizations
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Boca Raton, FL
CRC Press, Taylor & Francis Group
[2015]
|
| Schriftenreihe: | Monographs on statistics and applied probability
143 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028149296&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xv, 351 Seiten Illustrationen, Diagramme |
| ISBN: | 9781498712163 |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Preface XV
1 Introduction 1
2 The Lasso for Linear Models 7
2.1 Introduction 7
2.2 The Lasso Estimator 8
2.3 Cross-Validation and Inference 13
2.4 Computation of the Lasso Solution 14
2.4.1 Single Predictor: Soft Thresholding 15
2.4.2 Multiple Predictors: Cyclic Coordinate Descent 16
2.4.3 Soft-Thresholding and Orthogonal Bases 17
2.5 Degrees of Freedom 17
2.6 Uniqueness of the Lasso Solutions 19
2.7 A Glimpse at the Theory 20
2.8 The Nonnegative Garrote 20
2.9 £q Penalties and Bayes Estimates 22
2.10 Some Perspective 23
Exercises 24
3 Generalized Linear Models 29
3.1 Introduction 29
3.2 Logistic Regression 31
3.2.1 Example: Document Classification 32
3.2.2 Algorithms 35
3.3 Multiclass Logistic Regression 36
3.3.1 Example: Handwritten Digits 37
3.3.2 Algorithms 39
3.3.3 Grouped-Lasso Multinomial 39
3.4 Log-Linear Models and the Poisson GLM 40
3.4.1 Example: Distribution Smoothing 40
3.5 Cox Proportional Hazards Models 42
3.5.1 Cross-Validation 43
3.5.2 Pre-Validation 45
3.6 Support Vector Machines 46
3.6.1 Logistic Regression with Separable Data 49
IX
X
3.7 Computational Details and glmnet 50
Bibliographic Notes 52
Exercises 53
4 Generalizations of the Lasso Penalty 55
4.1 Introduction 55
4.2 The Elastic Net 56
4.3 The Group Lasso 58
4.3.1 Computation for the Group Lasso 62
4.3.2 Sparse Group Lasso 64
4.3.3 The Overlap Group Lasso 65
4.4 Sparse Additive Models and the Group Lasso 69
4.4.1 Additive Models and Backfitting 69
4.4.2 Sparse Additive Models and Backfitting 70
4.4.3 Approaches Using Optimization and the Group Lasso 72
4.4.4 Multiple Penalization for Sparse Additive Models 74
4.5 The Fused Lasso 76
4.5.1 Fitting the Fused Lasso 77
4.5.1.1 Reparametrization 78
4.5.1.2 A Path Algorithm 79
4.5.1.3 A Dual Path Algorithm 79
4.5.1.4 Dynamic Programming for the Fused Lasso 80
4.5.2 Trend Filtering 81
4.5.3 Nearly Isotonic Regression 83
4.6 Nonconvex Penalties 84
Bibliographic Notes 86
Exercises 88
5 Optimization Methods 95
5.1 Introduction 95
5.2 Convex Optimality Conditions 95
5.2.1 Optimality for Differentiable Problems 95
5.2.2 Nondifferentiable Functions and Subgradients 98
5.3 Gradient Descent 100
5.3.1 Unconstrained Gradient Descent 101
5.3.2 Projected Gradient Methods 102
5.3.3 Proximal Gradient Methods 103
5.3.4 Accelerated Gradient Methods 107
5.4 Coordinate Descent 109
5.4.1 Separability and Coordinate Descent 110
5.4.2 Linear Regression and the Lasso 112
5.4.3 Logistic Regression and Generalized Linear Models 115
5.5 A Simulation Study 117
5.6 Least Angle Regression 118
5.7 Alternating Direction Method of Multipliers 121
xi
5.8 Minorization-Maximization Algorithms 123
5.9 Biconvexity and Alternating Minimization 124
5.10 Screening Rules 127
Bibliographic Notes 131
Appendix 132
Exercises 134
6 Statistical Inference 139
6.1 The Bayesian Lasso 139
6.2 The Bootstrap 142
6.3 Post-Selection Inference for the Lasso 147
6.3.1 The Covariance Test 147
6.3.2 A General Scheme for Post-Selection Inference 150
6.3.2.1 Fixed-A Inference for the Lasso 154
6.3.2.2 The Spacing Test for LAR 156
6.3.3 What Hypothesis Is Being Tested? 157
6.3.4 Back to Forward Stepwise Regression 158
6.4 Inference via a Debiased Lasso 158
6.5 Other Proposals for Post-Selection Inference 160
Bibliographic Notes 161
Exercises 162
7 Matrix Decompositions, Approximations, and Completion 167
7.1 Introduction 167
7.2 The Singular Value Decomposition 169
7.3 Missing Data and Matrix Completion 169
7.3.1 The Netffix Movie Challenge 170
7.3.2 Matrix Completion Using Nuclear Norm 174
7.3.3 Theoretical Results for Matrix Completion 177
7.3.4 Maximum Margin Factorization and Related Methods 181
7.4 Reduced-Rank Regression 184
7.5 A General Matrix Regression Framework 185
7.6 Penalized Matrix Decomposition 187
7.7 Additive Matrix Decomposition 190
Bibliographic Notes 195
Exercises 196
8 Sparse Multivariate Methods 201
8.1 Introduction 201
8.2 Sparse Principal Components Analysis 202
8.2.1 Some Background 202
8.2.2 Sparse Principal Components 204
8.2.2.1 Sparsity from Maximum Variance 204
8.2.2.2 Methods Based on Reconstruction 206
Higher-Rank Solutions 207
8.2.3
8.2.3.1 Illustrative Application of Sparse PGA 209
8.2.4 Sparse PCA via Fantope Projection 210
8.2.5 Sparse Autoencoders and Deep Learning 210
8.2.6 Some Theory for Sparse PCA 212
8.3 Sparse Canonical Correlation Analysis 213
8.3.1 Example: Netflix Movie Rating Data 215
8.4 Sparse Linear Discriminant Analysis 217
8.4.1 Normal Theory and Bayes Rule 217
8.4.2 Nearest Shrunken Centroids 218
8.4.3 Fisher s Linear Discriminant Analysis 221
8.4.3.1 Example: Simulated Data with Five Classes 222
8.4.4 Optimal Scoring 225
8.4.4.1 Example: Face Silhouettes 226
8.5 Sparse Clustering 227
8.5.1 Some Background on Clustering 227
8.5.1.1 Example: Simulated Data with Six Classes 228
8.5.2 Sparse Hierarchical Clustering 228
8.5.3 Sparse /C-Means Clustering 230
8.5.4 Convex Clustering 231
Bibliographie Notes 232
Exercises 234
Graphs and Model Selection 241
9.1 Introduction 241
9.2 Basics of Graphical Models 241
9.2.1 Factorization and Markov Properties 241
9.2.1.1 Factorization Property 242
9.2.1.2 Markov Property 9.2.1.3 Equivalence of Factorization and Markov 243
Properties 243
9.2.2 Some Examples 244
9.2.2.1 Discrete Graphical Models 244
9.2.2.2 Gaussian Graphical Models 245
9.3 Graph Selection via Penalized Likelihood 246
9.3.1 Global Likelihoods for Gaussian Models 247
9.3.2 Graphical Lasso Algorithm 248
9.3.3 Exploiting Block-Diagonal Structure 251
9.3.4 Theoretical Guarantees for the Graphical Lasso 252
9.3.5 Global Likelihood for Discrete Models 253
9.4 Graph Selection via Conditional Inference 254
9.4.1 Neighborhood-Based Likelihood for Gaussians 255
9.4.2 Neighborhood-Based Likelihood for Discrete Models 256
9.4.3 Pseudo-Likelihood for Mixed Models 259
9.5 Graphical Models with Hidden Variables 261
Bibliographic Notes 261
Exercises
xiii
263
10 Signal Approximation and Compressed Sensing 269
10.1 Introduction 269
10.2 Signals and Sparse Representations 269
10.2.1 Orthogonal Bases 269
10.2.2 Approximation in Orthogonal Bases 271
10.2.3 Reconstruction in Overcomplete Bases 274
10.3 Random Projection and Approximation 276
10.3.1 Johnson-Lindenstrauss Approximation 277
10.3.2 Compressed Sensing 278
10.4 Equivalence between and t Recovery 280
10.4.1 Restricted Nullspace Property 281
10.4.2 Sufficient Conditions for Restricted Nullspace 282
10.4.3 Proofs 284
10.4.3.1 Proof of Theorem 10.1 284
10.4.3.2 Proof of Proposition 10.1 284
Bibliographic Notes 285
Exercises 286
11 Theoretical Results for the Lasso 289
11.1 Introduction 289
11.1.1 Types of Loss Functions 289
11.1.2 Types of Sparsity Models 290
11.2 Bounds on Lasso ¿2-Error 291
11.2.1 Strong Convexity in the Classical Setting 291
11.2.2 Restricted Eigenvalues for Regression 293
11.2.3 A Basic Consistency Result 294
11.3 Bounds on Prediction Error 299
11.4 Support Recovery in Linear Regression 301
11.4.1 Variable-Selection Consistency for the Lasso 301
11.4.1.1 Some Numerical Studies 303
11.4.2 Proof of Theorem 11.3 305
11.5 Beyond the Basic Lasso 309
Bibliographic Notes 311
Exercises 312
Bibliography 315
Author Index 337
Index
343
|
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| illustrated | Illustrated |
| indexdate | 2024-12-20T17:18:06Z |
| institution | BVB |
| isbn | 9781498712163 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028149296 |
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| publishDateSearch | 2015 |
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| series | Monographs on statistics and applied probability |
| series2 | Monographs on statistics and applied probability A Chapman & Hall book |
| spellingShingle | Hastie, Trevor 1953- Tibshirani, Robert 1956- Wainwright, Martin J. Statistical learning with sparsity the lasso and generalizations Monographs on statistics and applied probability Mathematical statistics Least squares Linear models (Statistics) Proof theory Statistik (DE-588)4056995-0 gnd Numerische Mathematik (DE-588)4042805-9 gnd Methode der kleinsten Quadrate (DE-588)4038974-1 gnd Schwach besetzte Matrix (DE-588)4056053-3 gnd Lineares Modell (DE-588)4134827-8 gnd Beweistheorie (DE-588)4145177-6 gnd |
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| title | Statistical learning with sparsity the lasso and generalizations |
| title_auth | Statistical learning with sparsity the lasso and generalizations |
| title_exact_search | Statistical learning with sparsity the lasso and generalizations |
| title_full | Statistical learning with sparsity the lasso and generalizations Trevor Hastie (Stanford University, USA), Robert Tibshirani (Stanford University, USA), Martin Wainwright (University of California, Berkeley, USA) |
| title_fullStr | Statistical learning with sparsity the lasso and generalizations Trevor Hastie (Stanford University, USA), Robert Tibshirani (Stanford University, USA), Martin Wainwright (University of California, Berkeley, USA) |
| title_full_unstemmed | Statistical learning with sparsity the lasso and generalizations Trevor Hastie (Stanford University, USA), Robert Tibshirani (Stanford University, USA), Martin Wainwright (University of California, Berkeley, USA) |
| title_short | Statistical learning with sparsity |
| title_sort | statistical learning with sparsity the lasso and generalizations |
| title_sub | the lasso and generalizations |
| topic | Mathematical statistics Least squares Linear models (Statistics) Proof theory Statistik (DE-588)4056995-0 gnd Numerische Mathematik (DE-588)4042805-9 gnd Methode der kleinsten Quadrate (DE-588)4038974-1 gnd Schwach besetzte Matrix (DE-588)4056053-3 gnd Lineares Modell (DE-588)4134827-8 gnd Beweistheorie (DE-588)4145177-6 gnd |
| topic_facet | Mathematical statistics Least squares Linear models (Statistics) Proof theory Statistik Numerische Mathematik Methode der kleinsten Quadrate Schwach besetzte Matrix Lineares Modell Beweistheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028149296&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002494005 |
| work_keys_str_mv | AT hastietrevor statisticallearningwithsparsitythelassoandgeneralizations AT tibshiranirobert statisticallearningwithsparsitythelassoandgeneralizations AT wainwrightmartinj statisticallearningwithsparsitythelassoandgeneralizations |