High-dimensional probability: an introduction with applications in data science
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
[2018]
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| Series: | Cambridge series in statistical and probabilistic mathematics
47 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 284 Seiten Diagramme |
| ISBN: | 9781108415194 |
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MARC
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Record in the Search Index
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| adam_text | Contents Foreword Preface page xi xiii Appetizer Using Probability to Cover a Geometric Set 0.1 Notes 1 4 1 1.1 1.2 1.3 1.4 Preliminaries on Random Variables Basic Quantities Associated with Random Variables Some Classical Inequalities Limit Theorems Notes 5 5 6 8 10 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Concentration of Sums of Independent Random Variables Why Concentration Inequalities? Hoeffding’s Inequality Chernoff’s Inequality Application: Degrees of Random Graphs Sub-Gaussian Distributions General Hoeffding and Khintchine Inequalities Sub-Exponential Distributions Bernstein’s Inequality Notes 11 11 13 17 19 21 26 28 33 36 3 3.1 3.2 3.3 3.4 3.5 38 39 41 45 51 3.6 3.7 3.8 Random Vectors in High Dimensions Concentration of the Norm Covariance Matrices and Principal Component Analysis Examples of High-Dimensional Distributions Sub-Gaussian Distributions in Higher Dimensions Application: Grothendieck’s Inequality and Semidefinite Programming Application: Maximum Cut for Graphs Kernel Trick, and Tightening of Grothendieck’s Inequality Notes 4 4.1 Random Matrices Preliminaries on Matrices 70 70 vii 55 60 64 68
Contents viii 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Nets, Covering Numbers, and Packing Numbers Application: Error Correcting Codes Upper Bounds on Random Sub֊Gaussian Matrices Application: Community Detection in Networks Two-Sided Bounds on Sub-Gaussian Matrices Application: Covariance Estimation and Clustering Notes 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Concentration Without Independence Concentration of Lipschitz Functions for the Sphere Concentration for Other Metric Measure Spaces Application: Johnson-Lindenstrauss Lemma Matrix Bernstein Inequality Application: Community Detection in Sparse Networks Application: Covariance Estimation for General Distributions Notes 98 98 104 110 113 121 122 125 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Quadratic Forms, Symmetrization, and Contraction Decoupling Hanson-Wright Inequality Concentration for Anisotropic Random Vectors Symmetrization Random Matrices With Non-I.I.D. Entries Application: Matrix Completion Contraction Principle Notes 127 127 130 134 136 138 140 143 145 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Random Processes Basic Concepts and Examples Slepian’s Inequality Sharp Bounds on Gaussian Matrices Sudakov’s Minoration Inequality Gaussian Width Stable Dimension, Stable Rank, and Gaussian Complexity Random Projections of Sets Notes 147 147 151 157 160 162 167 170 174 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Chaining Dudley’s Inequality Application: Empirical Processes VC Dimension Application: Statistical Learning Theory Generic Chaining Talagrand’s Majorizing Measure and Comparison Theorems Chevet’s Inequality Notes 176 176 183 188 200 206 210 212 214 75 79
83 87 91 93 97
ix Contents 9 9.1 9.2 9.3 9.4 9.5 Deviations of Random Matrices and Geometric Consequences Matrix Deviation Inequality Random Matrices, Random Projections, and Covariance Estimation The Johnson-Lindenstrauss Lemma for Infinite Sets Random Sections: Μ* Bound and Escape Theorem Notes 10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 Sparse Recovery High-Dimensional Signal Recovery Problems Signal Recovery Based on Μ* Bound Recovery of Sparse Signals Low-Rank Matrix Recovery Exact Recovery and the Restricted Isometry Property Lasso Algorithm for Sparse Regression Notes 11 11.1 11.2 11.3 11.4 Dvoretzky-Milman Theorem Deviations of Random Matrices with respect to General Norms Johnson-Lindenstrauss Embeddings and Sharper ChevetInequality Dvoretzky-Milman Theorem Notes Hints for Exercises References Index 216 216 222 225 227 231 232 232 234 236 239 241 247 252 254 254 257 259 264 265 272 281
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| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| indexdate | 2024-07-10T08:12:18Z |
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| isbn | 9781108415194 |
| language | English |
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| series | Cambridge series in statistical and probabilistic mathematics |
| series2 | Cambridge series in statistical and probabilistic mathematics |
| spelling | Vershynin, Roman 1974- (DE-588)1172311668 aut High-dimensional probability an introduction with applications in data science Roman Vershynin, University of California, Irvine Cambridge Cambridge University Press [2018] © 2018 xiv, 284 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge series in statistical and probabilistic mathematics 47 Zufallsvariable (DE-588)4129514-6 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Probabilities Stochastic processes Random variables Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Stochastischer Prozess (DE-588)4057630-9 s Zufallsvariable (DE-588)4129514-6 s DE-604 Cambridge series in statistical and probabilistic mathematics 47 (DE-604)BV011442366 47 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030621592&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vershynin, Roman 1974- High-dimensional probability an introduction with applications in data science Cambridge series in statistical and probabilistic mathematics Zufallsvariable (DE-588)4129514-6 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4129514-6 (DE-588)4064324-4 (DE-588)4057630-9 |
| title | High-dimensional probability an introduction with applications in data science |
| title_auth | High-dimensional probability an introduction with applications in data science |
| title_exact_search | High-dimensional probability an introduction with applications in data science |
| title_full | High-dimensional probability an introduction with applications in data science Roman Vershynin, University of California, Irvine |
| title_fullStr | High-dimensional probability an introduction with applications in data science Roman Vershynin, University of California, Irvine |
| title_full_unstemmed | High-dimensional probability an introduction with applications in data science Roman Vershynin, University of California, Irvine |
| title_short | High-dimensional probability |
| title_sort | high dimensional probability an introduction with applications in data science |
| title_sub | an introduction with applications in data science |
| topic | Zufallsvariable (DE-588)4129514-6 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Zufallsvariable Wahrscheinlichkeitsrechnung Stochastischer Prozess |
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