A mathematical introduction to compressive sensing
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Livre papier |
| Sprog: | Anglais |
| Udgivet: |
New York :
Birkhäuser : Springer
C 2013.
|
| Serier: | Applied and numerical harmonic analysis
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| Fag: | |
| Autres localisations: | Voir dans le Sudoc |
| Edition sous un autre format: | • A mathematical introduction to compressive sensing, by Simon Foucart, Holger Rauhut., 1st ed. 2013., New York, NY, Springer New York, 2013, Applied and Numerical Harmonic Analysis, 978-0-8176-4948-7 |
Indholdsfortegnelse:
- 1 An Invitation to Compressive Sensing
- 1.1 What is Compressive Sensing?
- 1.2 Applications, Motivations, and Extensions
- 1.3 Overview of the Book
- 2 Sparse Solutions of Underdetermined Systems
- 2.1 Sparsity and Compressibility
- 2.2 Minimal Number of Measurements
- 2.3 NP-Hardness of 0-Minimization
- 3 Basic Algorithms
- 3.1 Optimization Methods
- 3.2 Greedy Methods .
- 3.3 Thresholding-Based Methods
- 4 Basis Pursuit
- 4.1 Null Space Property
- 4.2 Stability
- 4.3 Robustness
- 4.4 Recovery of Individual Vectors
- 4.5 The Projected Cross-Polytope
- 4.6 Low-Rank Matrix Recovery
- 5 Coherence
- 5.1 Definitions and Basic Properties
- 5.2 Matrices with Small Coherence
- 5.3 Analysis of Orthogonal Matching Pursuit
- 5.4 Analysis of Basis Pursuit
- 5.5 Analysis of Thresholding Algorithms
- 6 Restricted Isometry Property
- 6.1 Definitions and Basic Properties
- 6.2 Analysis of Basis Pursuit
- 6.3 Analysis of Thresholding Algorithms
- 6.4 Analysis of Greedy Algorithms
- 7 Basic Tools from Probability Theory
- 7.1 Essentials from Probability
- 7.2 Moments and Tails
- 7.3 Cramer s Theorem and Hoeffding s Inequality
- 7.4 Subgaussian Random Variables
- 7.5 Bernstein Inequalities
- 8 Advanced Tools from Probability Theory
- 8.1 Expectation of Norms of Gaussian Vectors
- 8.2 Rademacher Sums and Symmetrization
- 8.3 Khintchine Inequalities
- 8.4 Decoupling
- 8.5 Noncommutative Bernstein Inequality
- 8.6 Dudley s Inequality
- 8.7 Slepian s and Gordon s Lemmas
- 8.8 Concentration of Measure
- 8.9 Bernstein Inequality for Suprema of Empirical Processes
- 9 Sparse Recovery with Random Matrices
- 9.1 Restricted Isometry Property for Subgaussian Matrices
- 9.2 Nonuniform Recovery
- 9.3 Restricted Isometry Property for Gaussian Matrices
- 9.4 Null Space Property for Gaussian Matrices
- 9.5 Relation to Johnson Lindenstrauss Embeddings
- 10 Gelfand Widths of 1-Balls
- 10.1 Definitions and Relation to Compressive Sensing
- 10.2 Estimate for the Gelfand Widths of 1-Balls
- 10.3 Applications to the Geometry of Banach Spaces
- 11 Instance Optimality and Quotient Property
- 11.1 Uniform Instance Optimality
- 11.2 Robustness and Quotient Property
- 11.3 Quotient Property for Random Matrices
- 11.4 Nonuniform Instance Optimality
- 12 Random Sampling in Bounded Orthonormal Systems
- 12.1 Bounded Orthonormal Systems
- 12.2 Uncertainty Principles and Lower Bounds
- 12.3 Nonuniform Recovery: Random Sign Patterns
- 12.4 Nonuniform Recovery: Deterministic Sign Patterns
- 12.5 Restricted Isometry Property
- 12.6 Discrete Bounded Orthonormal Systems
- 12.7 Relation to the 1-Problem
- 13 Lossless Expanders in Compressive Sensing
- 13.1 Definitions and Basic Properties
- 13.2 Existence of Lossless Expanders
- 13.3 Analysis of Basis Pursuit
- 13.4 Analysis of an Iterative Thresholding Algorithm
- 13.5 Analysis of a Simple Sublinear-Time Algorithm
- 14 Recovery of Random Signals using Deterministic Matrices
- 14.1 Conditioning of Random Submatrices
- 14.2 Sparse Recovery via 1-Minimization
- 15 Algorithms for 1-Minimization
- 5.1 The Homotopy Method
- 15.2 Chambolle and Pock s Primal-Dual Algorithm
- 15.3 Iteratively Reweighted Least Squares
- A Matrix Analysis
- A.1 Vector and Matrix Norms
- A.2 The Singular Value Decomposition
- A.3 Least Squares Problems
- A.4 Vandermonde Matrices
- A.5 Matrix Function
- B Convex Analysis
- B.1 Convex Sets
- B.2 Convex Functions
- B.3 The Convex Conjugate
- B.4 The Subdifferential
- B.5 Convex Optimization Problems
- B.6 Matrix Convexity
- C Miscellanea
- C.1 Fourier Analysis
- C.2 Covering Numbers
- C.3 The Gamma Function and Stirling s Formula
- C.4 The Multinomial Theorem