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A mathematical introduction to compressive sensing. (English) Zbl 1315.94002
Applied and Numerical Harmonic Analysis. New York, NY: Birkhäuser/Springer (ISBN 978-0-8176-4947-0/hbk; 978-0-8176-4948-7/ebook). xviii, 625 p. (2013).
There are two important features of this book. The first feature is the set of “Notes” sections presented at the end of each chapter. These Notes supplement the relevant references, historical remarks, additional facts and open questions studied in the respective chapters. They can be used to expand the students’ knowledge and understanding or as an opportunity to conduct a seminar. The second feature is the collection of exercises spread throughout the chapters, being an essential component for students to learn the presented stuff. Moreover, the book concludes with three appendices that cover basic material from matrix analysis, convex analysis, and other interesting topics that hint at advanced areas of mathematics.
The book contains fifteen chapters. The first chapter gives a brief description of compressive sensing with some motivations, applications and a detailed overview of the book. In Chapter 2 the notions of vector sparsity and compressibility with some basic algorithms are given. Also, the minimal number of linear measurements required to recover sparse vectors is investigated in two different settings. Finally, it is proved that \(\ell_0\)-minimization, the ideal recovery scheme, is NP-hard in general. In Chapter 3 a popular algorithm used in compressive sensing is presented. This algorithm covers three methods: optimization methods, greedy methods, and thresholding-based methods. Chapter 4 is focussed on the basis pursuit (\(\ell_1\)-minimization) strategy, which consists in solving a convex optimization problem. Necessary and sufficient conditions for exact or approximate reconstruction of the original sparse or compressible vector are discussed. Also, the recovery of sparse vectors via basis pursuit is interpreted in terms of polytope geometry. In Chapter 5 the notion of coherence of a matrix and some of its generalizations are introduced. The authors investigate how small the coherence can be and they give some matrices with small coherence. Finally, they discuss some sufficient conditions expressed in terms of the coherence that guarantee the success of orthogonal matching pursuit, basis pursuit, and thresholding algorithms. In Chapter 6 the concept of restricted isometry property is introduced, which is known as a uniform uncertainty principle. The success of sparse recovery is established under some conditions on restricted isometry constants for basis pursuit, thresholding-based algorithms and for greedy algorithms. Chapter 7 deals with the basic facts of probability. The relation of moments of random variables to their tails and deviation inequalities for sums of independent random variables by means of moment-generating functions are studied. Chapter 8 incorporates the expectation of the \(\ell_p\)-norm of a standard Gaussian vector for \(p = 1,2,\infty\). Some simple results for Rademacher sums as well as the symmetrization method are presented. Decoupling inequalities are introduced to replace one sequence of random variables in a double sum by an independent copy. The scalar Bernstein inequality for bounded random variables is extended to a powerful deviation inequality for the operator norm of sums of random matrices. Finally, a deviation inequality for the supremum of an empirical process, which is sometimes called Talagrand’s concentration inequality or Bousquet’s inequality, is discussed. The sparse recovery based on random matrices and some related topics are discussed in Chapters 9–12. In Chapter 13 another type of matrices (adjacency matrices of certain bipartite graphs) are introduced. These matrices, called lossless expanders, can be used when reconstructing sparse vectors from a limited number of measurements. It is proved that, using adjacency matrices as measurement matrices, a stable and robust reconstruction of sparse vectors via \(\ell_1\)-minimization can be obtained. Finally, the stability and robustness of a thresholding-based algorithm and a simple sublinear-time algorithm are analyzed. Chapter 14 deals with recovery of random sparse signals with deterministic matrices. The results of this chapter are weaker than the ones of Chapter 5 in the sense that they apply only to most signals instead of all signals, but they show that the deterministic bounds using coherence may be somewhat pessimistic even if no bounds on the restricted isometry constants are available. Also, bounds on the conditioning of a random column submatrix of a given matrix are derived. The \(\ell_1\)-minimization plays a key role as a recovery method for compressive sensing. Algorithms designed specifically for \(\ell_1\)-minimization may be faster than general-purpose methods. In Chapter 15 several of these algorithms are introduced and analyzed.
The picture of compressive sensing is the important feature of this book. The material presented here represents a solid foundation for the mathematical theory of compressive sensing.

MSC:
94-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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