A mathematical introduction to compressive sensing:
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Main Authors: | , |
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Format: | Book |
Language: | English |
Published: |
New York ; Heidelberg ; Dordrecht ; London
Birkhäuser
[2013]
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Series: | Applied and numerical harmonic analysis
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Subjects: | |
Online Access: | Inhaltsverzeichnis Klappentext |
Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
Physical Description: | 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. |
any_adam_object | 1 |
author | Foucart, Simon 1977- Rauhut, Holger 1974- |
author_GND | (DE-588)1041275412 (DE-588)1044609222 |
author_facet | Foucart, Simon 1977- Rauhut, Holger 1974- |
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classification_rvk | SK 470 SK 600 SK 950 SK 970 ST 284 |
classification_tum | ELT 517f MAT 420f |
ctrlnum | (OCoLC)862796696 (DE-599)BVBBV041100261 |
discipline | Informatik Elektrotechnik Mathematik |
format | Book |
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id | DE-604.BV041100261 |
illustrated | Illustrated |
indexdate | 2025-03-03T04:00:47Z |
institution | BVB |
isbn | 9780817649470 9781493900633 9780817649487 |
language | English |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026076662 |
oclc_num | 862796696 |
open_access_boolean | |
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physical | xviii, 625 Seiten Illustrationen, Diagramme |
publishDate | 2013 |
publishDateSearch | 2013 |
publishDateSort | 2013 |
publisher | Birkhäuser |
record_format | marc |
series2 | Applied and numerical harmonic analysis |
spellingShingle | Foucart, Simon 1977- Rauhut, Holger 1974- A mathematical introduction to compressive sensing Datenkompression (DE-588)4121121-2 gnd Mathematik (DE-588)4037944-9 gnd |
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 |
work_keys_str_mv | AT foucartsimon amathematicalintroductiontocompressivesensing AT rauhutholger amathematicalintroductiontocompressivesensing |
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2000 SK 450 F762 |
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