《高维概率：入门和数据科学中的应用》

橱窗的光

《高维概率：入门和数据科学中的应用》（2019）
pdf
作者Roman Vershynin
Contents
Preface vi
Appetizer: using probability to cover a geometric set 1
1 Preliminaries on random variables 6
1.1 Basic quantities associated with random variables 6
1.2 Some classical inequalities 7
1.3 Limit theorems 9
1.4 Notes 12
2 Concentration of sums of independent random variables 13
2.1 Why concentration inequalities? 13
2.2 Hoeding's inequality 16
2.3 Cherno's inequality 19
2.4 Application: degrees of random graphs 21
2.5 Sub-gaussian distributions 24
2.6 General Hoeding's and Khintchine's inequalities 29
2.7 Sub-exponential distributions 32
2.8 Bernstein's inequality 37
2.9 Notes 40
3 Random vectors in high dimensions 42
3.1 Concentration of the norm 43
3.2 Covariance matrices and principal component analysis 45
3.3 Examples of high-dimensional distributions 50
3.4 Sub-gaussian distributions in higher dimensions 56
3.5 Application: Grothendieck's inequality and semidenite programming 60
3.6 Application: Maximum cut for graphs 66
3.7 Kernel trick, and tightening of Grothendieck's inequality 70
3.8 Notes 74
4 Random matrices 76
4.1 Preliminaries on matrices 76
4.2 Nets, covering numbers and packing numbers 81
4.3 Application: error correcting codes 86
4.4 Upper bounds on random sub-gaussian matrices 89
4.5 Application: community detection in networks 93
4.6 Two-sided bounds on sub-gaussian matrices 97
4.7 Application: covariance estimation and clustering 99
4.8 Notes 103
5 Concentration without independence 105
5.1 Concentration of Lipschitz functions on the sphere 105
5.2 Concentration on other metric measure spaces 112
5.3 Application: Johnson-Lindenstrauss Lemma 118
5.4 Matrix Bernstein's inequality 121
5.5 Application: community detection in sparse networks 129
5.6 Application: covariance estimation for general distributions 129
5.7 Notes 133
6 Quadratic forms, symmetrization and contraction 135
6.1 Decoupling 135
6.2 Hanson-Wright Inequality 139
6.3 Concentration of anisotropic random vectors 142
6.4 Symmetrization 145
6.5 Random matrices with non-i.i.d. entries 147
6.6 Application: matrix completion 148
6.7 Contraction Principle 151
6.8 Notes 154
7 Random processes 156
7.1 Basic concepts and examples 156
7.2 Slepian's inequality 160
7.3 Sharp bounds on Gaussian matrices 167
7.4 Sudakov's minoration inequality 170
7.5 Gaussian width 172
7.6 Stable dimension, stable rank, and Gaussian complexity 178
7.7 Random projections of sets 181
7.8 Notes 185
8 Chaining 187
8.1 Dudley's inequality 187
8.2 Application: empirical processes 195
8.3 VC dimension 200
8.4 Application: statistical learning theory 212
8.5 Generic chaining 219
8.6 Talagrand's majorizing measure and comparison theorems 223
8.7 Chevet's inequality 225
8.8 Notes 227
9 Deviations of random matrices and geometric consequences 229
9.1 Matrix deviation inequality 229
9.2 Random matrices, random projections and covariance estimation 235
9.3 Johnson-Lindenstrauss Lemma for innite sets 238
9.4 Random sections: M bound and Escape Theorem 240
9.5 Notes 244
10 Sparse Recovery 246
10.1 High-dimensional signal recovery problems 246
10.2 Signal recovery based on M bound 248
10.3 Recovery of sparse signals 250
10.4 Low-rank matrix recovery 254
10.5 Exact recovery and the restricted isometry property 256
10.6 Lasso algorithm for sparse regression 262
10.7 Notes 267
11 Dvoretzky-Milman's Theorem 269
11.1 Deviations of random matrices with respect to general norms 269
11.2 Johnson-Lindenstrauss embeddings and sharper Chevet inequality 272
11.3 Dvoretzky-Milman's Theorem 274
11.4 Notes 279
Bibliography 280
Index 290

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