Verfügbarkeit: A mathematical introduction to compressive sensing

A mathematical introduction to compressive sensing:

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Bibliographische Detailangaben
Hauptverfasser:	Foucart, Simon 1977- (VerfasserIn), Rauhut, Holger 1974- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	New York ; Heidelberg ; Dordrecht ; London Birkhäuser [2013]
Schriftenreihe:	Applied and numerical harmonic analysis
Schlagworte:	Datenkompression Mathematik
Online-Zugang:	Inhaltsverzeichnis Klappentext
Beschreibung:	Hier auch später erschienene, unveränderte Nachdrucke
Beschreibung:	xviii, 625 Seiten Illustrationen, Diagramme
ISBN:	9780817649470 9781493900633 9780817649487

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adam_text	Contents An Invitation to Compressive Sensing .................................... 1 1.1 What is Compressive Sensing? ...................................... 1 1.2 Applications, Motivations, and Extensions ......................... 8 1.3 Overview of the Book ............................................... 23 Notes .......................................................................... 33 Sparse Solutions of Underdetermined Systems ......................... 41 2.1 Sparsity and Compressibility ........................................ 41 2.2 Minimal Number of Measurements ................................. 48 2.3 NP-Hardness of £0-Minimization ................................... 53 Notes .......................................................................... 56 Exercises ..................................................................... 57 Basic Algorithms ............................................................ 61 3.1 Optimization Methods ............................................... 61 3.2 Greedy Methods ..................................................... 65 3.3 Thresholding-Based Methods ....................................... 69 Notes .......................................................................... 71 Exercises ..................................................................... 73 Basis Pursuit ................................................................ 77 4.1 Null Space Property ................................................. 78 4.2 Stability .............................................................. 82 4.3 Robustness ........................................................... 85 4.4 Recovery of Individual Vectors ..................................... 90 4.5 The Projected Cross-Polytope ....................................... 98 4.6 Low-Rank Matrix Recovery ......................................... 102 Notes .......................................................................... 104 Exercises ..................................................................... 105 Coherence ................................................................... Ill 5.1 Definitions and Basic Properties .................................... Ill 5.2 Matrices with Small Coherence ..................................... 113 XV xvi Contents 5.3 Analysis of Orthogonal Matching Pursuit .......................... 123 5.4 Analysis of Basis Pursuit ............................................ 124 5.5 Analysis of Thresholding Algorithms .............................. 126 Notes .......................................................................... 128 Exercises ..................................................................... 129 6 Restricted Isometry Property ............................................. 133 6.1 Definitions and Basic Properties .................................... 133 6.2 Analysis of Basis Pursuit ............................................ 141 6.3 Analysis of Thresholding Algorithms .............................. 148 6.4 Analysis of Greedy Algorithms ..................................... 156 Notes .......................................................................... 168 Exercises ..................................................................... 170 7 Basic Tools from Probability Theory ..................................... 175 7.1 Essentials from Probability .......................................... 175 7.2 Moments and Tails ................................................... 185 7.3 Cramer s Theorem and Hoeffding s Inequality ..................... 188 7.4 Subgaussian Random Variables ..................................... 191 7.5 Bernstein Inequalities ................................................ 196 Notes .......................................................................... 198 Exercises ..................................................................... 199 8 Advanced Tools from Probability Theory ............................... 201 8.1 Expectation of Norms of Gaussian Vectors ......................... 202 8.2 Rademacher Sums and Symmetrization ............................ 205 8.3 Khintchine Inequalities .............................................. 206 8.4 Decoupling ........................................................... 211 8.5 Noncommutative Bernstein Inequality .............................. 217 8.6 Dudley s Inequality .................................................. 224 8.7 Slepian s and Gordon s Lemmas .................................... 227 8.8 Concentration of Measure ........................................... 237 8.9 Bernstein Inequality for Suprema of Empirical Processes ......... 247 Notes .......................................................................... 260 Exercises ..................................................................... 265 9 Sparse Recovery with Random Matrices ................................ 271 9.1 Restricted Isometry Property for Subgaussian Matrices ........... 272 9.2 Nonuniform Recovery ............................................... 281 9.3 Restricted I some try Property for Gaussian Matrices ............... 291 9.4 Null Space Property for Gaussian Matrices ........................ 293 9.5 Relation to Johnson—Lindenstrauss Embeddings ................... 298 Notes .......................................................................... 302 Exercises ..................................................................... 306 Contents xvu 10 Gelfand Widths of ¿ι -Balls ................................................ 311 10.1 Definitions and Relation to Compressive Sensing ................. 311 10.2 Estimate for the Gelfand Widths of ¿i-Balls ....................... 317 10.3 Applications to the Geometry of Banach Spaces ................... 322 Notes .......................................................................... 327 Exercises ..................................................................... 328 11 Instance Optimality and Quotient Property ............................. 331 11.1 Uniform Instance Optimality ........................................ 332 11.2 Robustness and Quotient Property .................................. 337 11.3 Quotient Property for Random Matrices ............................ 343 11.4 Nonuniform Instance Optimality .................................... 358 Notes .......................................................................... 362 Exercises ..................................................................... 363 12 Random Sampling in Bounded Orthonormal Systems ................. 367 12.1 Bounded Orthonormal Systems ..................................... 368 12.2 Uncertainty Principles and Lower Bounds ......................... 374 12.3 Nonuniform Recovery: Random Sign Patterns ..................... 384 12.4 Nonuniform Recovery: Deterministic Sign Patterns ............... 392 12.5 Restricted Isometry Property ........................................ 404 12.6 Discrete Bounded Orthonormal Systems ........................... 416 12.7 Relation to the /Ц-РгоМет ......................................... 418 Notes .......................................................................... 420 Exercises ..................................................................... 431 13 Lossless Expanders in Compressive Sensing ............................ 435 13.1 Definitions and Basic Properties .................................... 435 13.2 Existence of Lossless Expanders .................................... 439 13.3 Analysis of Basis Pursuit ............................................ 443 13.4 Analysis of an Iterative Thresholding Algorithm .................. 447 13.5 Analysis of a Simple Sublinear- Time Algorithm ................... 452 Notes .......................................................................... 455 Exercises ..................................................................... 456 14 Recovery of Random Signals using Deterministic Matrices ........... 459 14.1 Conditioning of Random Submatrices .............................. 460 14.2 Sparse Recovery via £i -Minimization .............................. 469 Notes .......................................................................... 471 Exercises ..................................................................... 473 15 Algorithms for £±-Minimization .......................................... 475 15.1 The Homotopy Method .............................................. 476 15.2 Chambolle and Pock s Primal-Dual Algorithm .................... 481 15.3 Iteratively Reweighted Least Squares ............................... 492 Notes .......................................................................... 503 Exercises ..................................................................... 511 xviii Contents Appendix.......................................................................... 515 A Matrix Analysis............................................................. 515 Α. 1 Vector and Matrix Norms ............................................ 515 A.2 The Singular Value Decomposition ................................. 525 A.3 Least Squares Problems ............................................. 531 A.4 Vandermonde Matrices .............................................. 535 A.5 Matrix Functions ..................................................... 537 В Convex Analysis ............................................................ 543 B.I Convex Sets .......................................................... 543 B.2 Convex Functions .................................................... 545 B.3 The Convex Conjugate .............................................. 549 B.4 The Subdifferential .................................................. 551 B.5 Convex Optimization Problems ..................................... 555 B.6 Matrix Convexity .................................................... 564 С Miscellanea .................................................................. 573 C. 1 Fourier Analysis ..................................................... 573 C.2 Covering Numbers ................................................... 575 C.3 The Gamma Function and Stirling s Formula ...................... 577 C.4 The Multinomial Theorem .......................................... 580 C.5 Some Elementary Estimates ......................................... 580 C.6 Estimates of Some Integrals ......................................... 581 C.7 Hahn-Banach Theorems ............................................ 583 C.8 Smoothing Lipschitz Functions ..................................... 583 C.9 Weak and Distributional Derivatives ................................ 585 C. 10 Differential Inequalities ............................................. 587 List of Symbols ................................................................... 589 References ......................................................................... 593 Index ............................................................................... 621 Simon Fou са rt, Holger Ra ii h ii t A Mathematical Introduction to Compressive Sensing At the intersection of mathemat ics, engineering, and louipulcr science sits lhe ι Iirivini»; field of compressive sensi ni* . Hasedon lhe premise lhal data accjiíisilion and compression can he performed simultaneously , compressive sensing finds applications in і таці n^. signal processing, and iiianx other domains. In the arcas oí applied mathemaiics, electrical engineering, and theoretical computer science, an explosion о 1 research act iv ft has already Toll owed t he í heoret ical results that highlighted the efficiency of í he basic principles. The ele^a nt ideas heh i ud l líese pri nciples are also of independent interest to pure mathematicians. Л fat/icììuiticai Introduction t o ( . o/nprcssiì c Sensing l^îvcn a detailed account of (lie core t heor upon w lì ic li the field is build. Ke feat u res include: lhe first textbook com pi et el devoted to the topic о Г coni press i ve sensing ( omprehen si e treatment of the subject, including background material from probabii i t tlicoľx, detailed proof s ol t he main theorems, a nel an out I ine of possi ble appi ica t ions Numerous exercises designed to help si nelení s u iidersl and t he material - n exlen sive bibi iòirraphx w ith o er ^(X) references t hal u*u ide resen rehers t hnniirh t he literal li re Wil h onh moderale ¡jrerècfuisïtes, A Siaiłieiiiatii al Introduction to Compressive Scusili^ is an excellent textbocik for graduate courses in mat hematics, engineering;, and eoniputer science. It also serves as a reliable resource for practitioners and researchers in these disciplines who want to accju і re a careful understanding of the subject.
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subject_GND	(DE-588)4121121-2 (DE-588)4037944-9
title	A mathematical introduction to compressive sensing
title_auth	A mathematical introduction to compressive sensing
title_exact_search	A mathematical introduction to compressive sensing
title_full	A mathematical introduction to compressive sensing Simon Foucart ; Holger Rauhut
title_fullStr	A mathematical introduction to compressive sensing Simon Foucart ; Holger Rauhut
title_full_unstemmed	A mathematical introduction to compressive sensing Simon Foucart ; Holger Rauhut
title_short	A mathematical introduction to compressive sensing
title_sort	a mathematical introduction to compressive sensing
topic	Datenkompression (DE-588)4121121-2 gnd Mathematik (DE-588)4037944-9 gnd
topic_facet	Datenkompression Mathematik
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026076662&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026076662&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
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