High-dimensional statistics: a non-asymptotic viewpoint
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge, United Kingdom ; New York, NY, USA
Cambridge University Press
2019
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| Schriftenreihe: | Cambridge series in statistical and probabilistic mathematics
48 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | xvii, 552 Seiten Illustrationen, Diagramme |
| ISBN: | 9781108498029 |
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| 245 | 1 | 0 | |a High-dimensional statistics |b a non-asymptotic viewpoint |c Martin J. Wainwright (University of California, Berkeley) |
| 264 | 1 | |a Cambridge, United Kingdom ; New York, NY, USA |b Cambridge University Press |c 2019 | |
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Datensatz im Suchindex
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List of chapters List of illustrations xv 1 1 Introduction 2 Basic tail and concentration bounds 21 3 Concentration of measure 58 4 Uniform laws of large numbers 98 5 Metric entropy and its uses 121 6 Random matrices and covariance estimation 159 7 Sparse linear models in high dimensions 194 8 Principal component analysis in high dimensions 236 9 Decomposability and restricted strong convexity 259 10 Matrix estimation with rank constraints 312 11 Graphical models for high-dimensional data 347 12 Reproducing kernel Hilbert spaces 383 13 Nonparametric least squares 416 14 Localization and uniform laws 453 15 Minimax lower bounds References Subject index Author index 485 524 540 548 vii
Contents List of illustrations xv 1 Introduction 1.1 1.2 1.6 Classical versus high-dimensional theory What can go wrong in high dimensions? 1.2.1 Linear discriminant analysis 1.2.2 Covariance estimation 1.2.3 Nonparametric regression What can help us in high dimensions? 1.3.1 Sparsity in vectors 1.3.2 Structure in covariance matrices 1.3.3 Structured forms of regression What is the non-asymptotic viewpoint? Overview of the book 1.5.1 Chapter structure and synopses 1.5.2 Recommended background 1.5.3 Teaching possibilities and a flow diagram Bibliographic details and background 1 1 2 2 5 7 9 10 11 12 14 15 15 17 17 19 2 Basic tail and concentration bounds 2.1 2.3 2.4 2.5 2.6 2.7 Classical bounds 2.1.1 From Markov to Chernoff 2.1.2 Sub-Gaussian variables and Hoeffding bounds 2.1.3 Sub-exponential variables and Bernstein bounds 2.1.4 Some one-sided results Martingale-based methods 2.2.1 Background 2.2.2 Concentration bounds for martingale difference sequences Lipschitz functions of Gaussian variables Appendix A: Equivalent versions of sub-Gaussian variables Appendix B: Equivalent versions of sub-exponential variables Bibliographic details and background Exercises 3 Concentration of measure 3.1 Concentration by entropie techniques 3.1.1 Entropy and its properties 3.1.2 Herbst argument and its extensions 1.3 1.4 1.5 2.2 21 21 21 22 25 31 32 33 35 40 45 48 49 50 58 58 58 60 ix
Contents x 3.1.3 3.1.4 3.2 A geometric perspective on concentration 3.2.1 3.2.2 3.2.3 3.3 Concentration functions Connection to Lipschitz functions From geometry to concentration Wasserstein distances and information inequalities 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4 Separately convex functions and the entropie method Tensorization and separately convex functions Wasserstein distances Transportation cost and concentration inequalities Tensorization for transportation cost Transportation cost inequalities for Markov chains Asymmetric coupling cost Tail bounds for empirical processes 3.4.1 3.4.2 A functional Hoeffding inequality A functional Bernstein inequality 3.5 3.6 Bibliographic details and background Exercises 4 4.1 Uniform laws of large numbers Motivation 4.1.1 4.1.2 4.2 A uniform law via Rademacher complexity 4.2.1 4.3 Uniform convergence of cumulative distribution functions Uniform laws for more general function classes Necessary conditions with Rademacher complexity Upper bounds on the Rademacher complexity 4.3.1 4.3.2 4.3.3 Classes with polynomial discrimination Vapnik-Chervonenkis dimension Controlling the VC dimension 4.4 4.5 Bibliographic details and background Exercises 5 5.1 5.2 5.3 Metric entropy and its uses Covering and packing Gaussian and Rademacher complexity Metric entropy and sub-Gaussian processes 62 64 67 67 70 72 76 76 78 80 82 84 87 87 89 91 92 98 98 98 101 104 107 109 109 111 115 117 117 5.5 5.6 5.7 5.8 Sudakov’s lower bound Chaining and Orlicz processes Bibliographic details and background Exercises 121 121 132 134 135 137 139 143 143 145 146 148 150
153 154 6 6.1 Random matrices and covariance estimation Some preliminaries 159 159 5.3.1 5.3.2 5.3.3 5.4 Upper bound by one-step discretization Some examples of discretization bounds Chaining and Dudley’s entropy integral Some Gaussian comparison inequalities 5.4.1 5.4.2 5.4.3 A general comparison result Slepian and Sudakov-Fernique inequalities Gaussian contraction inequality
Contents 6.1.1 6.1.2 6.2 6.3 6.4 Wishart matrices and their behavior Covariance matrices from sub-Gaussian ensembles Bounds for general matrices 6.4.1 6.4.2 6.4.3 6.4.4 6.4.5 6.5 Notation and basic facts Set-up of covariance estimation Background on matrix analysis Tail conditions for matrices Matrix Chernoff approach and independent decompositions Upper tail bounds for random matrices Consequences for covariance matrices Bounds for structured covariance matrices 6.5.1 6.5.2 Unknown sparsity and thresholding Approximate sparsity 6.6 6.7 6.8 Appendix: Proof of Theorem 6.1 Bibliographic details and background Exercises 7 7.1 Sparse linear models in high dimensions Problem formulation and applications 7.1.1 7.1.2 7.2 Recovery in the noiseless setting 7.2.1 7.2.2 7.2.3 7.3 Λ -based relaxation Exact recovery and restricted nullspace Sufficient conditions for restricted nullspace Estimation in noisy settings 7.3.1 7.3.2 7.3.3 7.4 7.5 Different sparsity models Applications of sparse linear models Restricted eigenvalue condition Bounds on քշ-error for hard sparse models Restricted nullspace and eigenvalues for random designs Bounds on prediction error Variable or subset selection 7.5.1 7.5.2 Variable selection consistency for the Lasso Proof of Theorem 7.21 7.6 7.7 7.8 Appendix: Proof of Theorem 7.16 Bibliographic details and background Exercises 8 8.1 Principal component analysis in high dimensions Principal components and dimension reduction 8.1.1 8.1.2 8.2 Bounds for generic eigenvectors 8.2.1 8.2.2 8.3 A general deterministic result Consequences for a spiked ensemble Sparse
principal component analysis 8.3.1 8.3.2 8.4 8.5 Interpretations and uses of PCA Perturbations of eigenvalues and eigenspaces A general deterministic result Consequences for the spiked model with sparsity Bibliographic details and background Exercises xi 159 160 161 165 168 168 169 172 174 179 180 180 183 185 188 189 194 194 194 196 199 200 200 202 206 207 209 213 216 218 219 222 224 227 229 236 236 237 241 242 242 245 248 249 252 255 256
Contents xii 9 9.1 9.2 Decomposability and restricted strong convexity A general regularized Aí-estímator Decomposable regularizes and their utility 9.2.1 9.2.2 9.3 Restricted curvature conditions 9.3.1 9.4 Guarantees under restricted strong convexity Bounds under Փ*-curvature Bounds for sparse vector regression 9.5.1 9.5.2 9.5.3 9.6 9.7 9.8 Restricted strong convexity Some general theorems 9.4.1 9.4.2 9.5 Definition and some examples A key consequence of decomposability Generalized linear models with sparsity Bounds under restricted strong convexity Bounds under ¿„-curvature conditions Bounds for group-structured sparsity Bounds for overlapping decomposition-based norms Techniques for proving restricted strong convexity 9.8.1 9.8.2 Lipschitz cost functions and Rademacher complexity A one-sided bound via truncation 9.9 9.10 9.11 Appendix: Star-shaped property Bibliographic details and background Exercises 10 10.1 10.2 Matrix estimation with rank constraints Matrix regression and applications Analysis of nuclear norm regularization 10.2.1 Decomposability and subspaces 10.2.2 Restricted strong convexity and error bounds 10.2.3 Bounds under operator norm curvature 10.3 10.4 10.5 10.6 10.7 10.8 10.9 Matrix compressed sensing Bounds for phase retrieval Multivariate regression with low-rank constraints Matrix completion Additive matrix decompositions Bibliographic details and background Exercises 11 11.1 Graphical models for high-dimensional data Some basics 11.1.1 11.1.2 11.1.3 11.1.4 11.2 Factorization Conditional independence Hammersley-Cliiford equivalence Estimation of
graphical models Estimation of Gaussian graphical models 11.2.1 Graphical Lasso: ¿i -regularized maximum likelihood 11.2.2 Neighborhood-based methods 11.3 Graphical models in exponential form 11.3.1 A general form of neighborhood regression 11.3.2 Graph selection for Ising models 259 259 269 269 272 276 277 279 280 284 286 286 287 288 290 293 297 298 302 306 306 307 312 312 317 317 319 320 321 326 329 330 337 341 343 347 347 347 350 351 352 352 353 359 365 366 367
Contents 11.4 Graphs with corrupted or hidden variables 11.4.1 Gaussian graph estimation with corrupted data 11.4.2 Gaussian graph selection with hidden variables 11.5 11.6 Bibliographic details and background Exercises 12 12.1 12.2 Reproducing kernel Hilbert spaces Basics of Hilbert spaces Reproducing kernel Hilbert spaces 12.2.1 12.2.2 12.2.3 12.2.4 12.3 12.4 Positive semidefinite kernel functions Feature maps in ¿2(N) Constructing an RKHS from a kernel A more abstract viewpoint and further examples Mercer’s theorem and its consequences Operations on reproducing kernel Hilbert spaces 12.4.1 Sums of reproducing kernels 12.4.2 Tensor products 12.5 Interpolation and fitting 12.5.1 Function interpolation 12.5.2 Fitting via kernel ridge regression 12.6 12.7 12.8 Distances between probability measures Bibliographic details and background Exercises 13 13.1 Nonparametric least squares Problem set-up 13.1.1 Different measures of quality 13.1.2 Estimation via constrained least squares 13.1.3 Some examples 13.2 Bounding the prediction error 13.2.1 13.2.2 13.2.3 13.2.4 13.3 Bounds via metric entropy Bounds for high-dimensional parametric problems Bounds for nonparametric problems Proof of Theorem 13.5 Oracle inequalities 13.3.1 Some examples of oracle inequalities 13.3.2 Proof of Theorem 13.13 13.4 Regularized estimators 13.4.1 13.4.2 13.4.3 13.4.4 Oracle inequalities for regularized estimators Consequences for kernel ridge regression Proof of Corollary 13.18 Proof of Theorem 13.17 13.5 13.6 Bibliographic details and background Exercises 14 Localization and uniform laws Population
and empirical ¿2-norms 14.1 14.1.1 A uniform law with localization 14.1.2 Specialization to kernel classes xiii 368 368 373 376 378 383 383 385 386 387 388 390 394 400 400 403 405 405 407 409 411 412 416 416 416 417 418 420 425 427 429 430 432 434 437 439 439 439 443 444 448 449 453 453 454 458
xiv Contents 14.1.3 Proof of Theorem 14.1 14.2 A one-sided uniform law 14.2.1 Consequences for nonparametric least squares 14.2.2 Proof of Theorem 14.12 14.3 A uniform law for Lipschitz cost functions 14.3.1 General prediction problems 14.3.2 Uniform law for Lipschitz cost functions 14.4 Some consequences for nonparametric density estimation 14.4.1 Density estimation via the nonparametric maximum likelihood estimate 14.4.2 Density estimation via projections 14.5 14.6 14.7 Appendix: Population and empirical Rademacher complexities Bibliographic details and background Exercises 15 15.1 Minimax lower bounds Basic framework 15.1.1 Minimax risks 15.1.2 From estimation to testing 15.1.3 Some divergence measures 15.2 Binary testing and Le Cam’s method 15.2.1 Bayes error and total variation distance 15.2.2 Le Cam’s convex hull method 15.3 Fano’s method 15.3.1 15.3.2 15.3.3 15.3.4 15.3.5 Kullback-Leibler divergence and mutual information Fano lower bound on minimax risk Bounds based on local packings Local packings with Gaussian entropy bounds Yang-Barron version of Fano’s method 15.4 Appendix: Basic background in information theory 15.5 Bibliographic details and background 15.6 Exercises References Subject index Author index 460 462 466 468 469 469 472 475 475 477 480 481 482 485 485 486 487 489 491 491 497 500 501 501 503 506 512 515 518 519 524 540 548 |
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| spelling | Wainwright, Martin J. Verfasser (DE-588)1020083271 aut High-dimensional statistics a non-asymptotic viewpoint Martin J. Wainwright (University of California, Berkeley) Cambridge, United Kingdom ; New York, NY, USA Cambridge University Press 2019 xvii, 552 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge series in statistical and probabilistic mathematics 48 Hier auch später erschienene, unveränderte Nachdrucke Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Maschinelles Lernen (DE-588)4193754-5 gnd rswk-swf Statistik (DE-588)4056995-0 gnd rswk-swf Mathematical statistics / Textbooks Big data Mathematical statistics Textbooks Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Statistik (DE-588)4056995-0 s Maschinelles Lernen (DE-588)4193754-5 s DE-604 Erscheint auch als Online-Ausgabe 978-1-108-62777-1 (DE-604)BV045485616 Cambridge series in statistical and probabilistic mathematics 48 (DE-604)BV011442366 48 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=030882193&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wainwright, Martin J. High-dimensional statistics a non-asymptotic viewpoint Cambridge series in statistical and probabilistic mathematics Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Maschinelles Lernen (DE-588)4193754-5 gnd Statistik (DE-588)4056995-0 gnd |
| subject_GND | (DE-588)4079013-7 (DE-588)4193754-5 (DE-588)4056995-0 |
| title | High-dimensional statistics a non-asymptotic viewpoint |
| title_auth | High-dimensional statistics a non-asymptotic viewpoint |
| title_exact_search | High-dimensional statistics a non-asymptotic viewpoint |
| title_full | High-dimensional statistics a non-asymptotic viewpoint Martin J. Wainwright (University of California, Berkeley) |
| title_fullStr | High-dimensional statistics a non-asymptotic viewpoint Martin J. Wainwright (University of California, Berkeley) |
| title_full_unstemmed | High-dimensional statistics a non-asymptotic viewpoint Martin J. Wainwright (University of California, Berkeley) |
| title_short | High-dimensional statistics |
| title_sort | high dimensional statistics a non asymptotic viewpoint |
| title_sub | a non-asymptotic viewpoint |
| topic | Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Maschinelles Lernen (DE-588)4193754-5 gnd Statistik (DE-588)4056995-0 gnd |
| topic_facet | Wahrscheinlichkeitstheorie Maschinelles Lernen Statistik |
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