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A Mathematical Introduction to Compressive Sensing.pdf

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    ANHA Series Preface
    Preface
    Contents
    Chapter 1 An Invitation to Compressive Sensing
    1.1 What is Compressive Sensing?
    1.2 Applications, Motivations, and Extensions
    1.3 Overview of the Book
    Notes
    Chapter 2 Sparse Solutions of Underdetermined Systems
    2.1 Sparsity and Compressibility
    2.2 Minimal Number of Measurements
    2.3 NP-Hardness of 0-Minimization
    Notes
    Exercises
    Chapter 3 Basic Algorithms
    3.1 Optimization Methods
    3.2 Greedy Methods
    3.3 Thresholding-Based Methods
    Notes
    Exercises
    Chapter 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
    Notes
    Exercises
    Chapter 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
    Notes
    Exercises
    Chapter 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
    Notes
    Exercises
    Chapter 7 Basic Tools from Probability Theory
    7.1 Essentials from Probability
    7.2 Moments and Tails
    7.3 Cramér's Theorem and Hoeffding's Inequality
    7.4 Subgaussian Random Variables
    7.5 Bernstein Inequalities
    Notes
    Exercises
    Chapter 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
    Notes
    Exercises
    Chapter 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
    Notes
    Exercises
    Chapter 10 Gelfand Widths of ℓ1-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
    Notes
    Exercises
    Chapter 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
    Notes
    Exercises
    Chapter 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 Λ1-Problem
    Notes
    Exercises
    Chapter 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
    Notes
    Exercises
    Chapter 14 Recovery of Random Signals using Deterministic Matrices
    14.1 Conditioning of Random Submatrices
    14.2 Sparse Recovery via 1-Minimization
    Notes
    Exercises
    Chapter 15 Algorithms for ℓ1-Minimization
    15.1 The Homotopy Method
    15.2 Chambolle and Pock's Primal-Dual Algorithm
    15.3 Iteratively Reweighted Least Squares
    Notes
    Exercises
    Appendix 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 Functions
    Appendix 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
    Appendix C Miscellanea
    C.1 Fourier Analysis
    C.2 Covering Numbers
    C.3 The Gamma Function and Stirling's Formula
    C.4 The Multinomial Theorem
    C.5 Some Elementary Estimates
    C.6 Estimates of Some Integrals
    C.7 Hahn–Banach Theorems
    C.8 Smoothing Lipschitz Functions
    C.9 Weak and Distributional Derivatives
    C.10 Differential Inequalities
    List of Symbols
    References
    Applied and Numerical Harmonic Analysis (63 Volumes)
    Index
    Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland College Park, MD, USA Editorial Advisory Board Akram Aldroubi Vanderbilt University Nashville, TN, USA Andrea Bertozzi University of California Los Angeles, CA, USA Douglas Cochran Arizona State University Phoenix, AZ, USA Hans G. Feichtinger University of Vienna Vienna, Austria Jelena Kovaˇcevi´c Carnegie Mellon University Pittsburgh, PA, USA Gitta Kutyniok Technische Universit¨at Berlin Berlin, Germany Mauro Maggioni Duke University Durham, NC, USA Zuowei Shen National University of Singapore Singapore, Singapore Christopher Heil Georgia Institute of Technology Atlanta, GA, USA Thomas Strohmer University of California Davis, CA, USA St´ephane Jaffard University of Paris XII Paris, France Yang Wang Michigan State University East Lansing, MI, USA For further volumes: http://www.springer.com/series/4968
    Simon Foucart Holger Rauhut A Mathematical Introduction to Compressive Sensing
    Simon Foucart Department of Mathematics Drexel University Philadelphia, PA, USA Holger Rauhut Lehrstuhl C f¨ur Mathematik (Analysis) RWTH Aachen University Aachen, Germany ISBN 978-0-8176-4947-0 DOI 10.1007/978-0-8176-4948-7 Springer New York Heidelberg Dordrecht London ISBN 978-0-8176-4948-7 (eBook) Library of Congress Control Number: 2013939591 Mathematics Subject Classification (2010): 46B09, 68P30, 90C90, 94A08, 94A12, 94A20 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)
    ANHA Series Preface The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reflects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flour- ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as signal processing, partial differential equations (PDEs), and image processing is reflected in our state-of-the-art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time–frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them. For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theory depends not only on classical Fourier analysis, but also on ideas from abstract harmonic analysis, including von Neumann algebras and the affine group. This leads to a study of the Heisenberg group and its relationship to Gabor systems, and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying influence of wavelet theory in the aforementioned topics illustrates the justification for providing a means for centralizing and disseminating information from the broader, but still focused, area of harmonic analysis. This will be a key role of ANHA. We intend to publish the scope and interaction that such a host of issues demands. v
    vi ANHA Series Preface Along with our commitment to publish mathematically significant works at the frontiers of harmonic analysis, we have a comparably strong commitment to publish major advances in the following applicable topics in which harmonic analysis plays a substantial role: Antenna theory Biomedical signal processing Digital signal processing Fast algorithms Gabor theory and applications Image processing Numerical partial differential equations Prediction theory Radar applications Sampling theory Spectral estimation Speech processing Time–frequency and time-scale analysis Wavelet theory The above point of view for the ANHA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many fields. In the last two centuries, Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientific phenomena, and on the solution of some of the most important problems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were defined, e.g., the concept of “function”. Since the coefficients of Fourier series are integrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set theory was also developed because of such uniqueness questions. A basic problem in Fourier analysis is to show how complicated phenomena, such as sound waves, can be described in terms of elementary harmonics. There are two aspects of this problem: first, to find, or even define properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second, to determine which phenomena can be constructed from given classes of harmonics, as done, e.g., by the mechanical synthesizers in tidal analysis. Fourier analysis is also the natural setting for many other problems in engineer- ing, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourier analysis not only characterizes the behavior of the prime numbers, but also provides the proper notion of spectrum for phenomena such as white light; this latter process leads to the Fourier analysis associated with correlation functions in filtering and prediction problems, and these problems, in turn, deal naturally with Hardy spaces in the theory of complex variables. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or filter design, or
    ANHA Series Preface vii the adaptive modeling inherent in time–frequency-scale methods such as wavelet theory. The coherent states of mathematical physics are translated and modulated Fourier transforms, and these are used, in conjunctionwith the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raison d’ˆetre of the ANHA series! University of Maryland College Park John J. Benedetto Series Editor
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