Book Review: A mathematical introduction to compressive sensing

Abstract

A mathematical introduction to compressive sensing, by S. Foucart and H. Rauhut, Applied and Numeric Harmonic Analysis, Birkhäuser/Springer, New York, 2013, xviii+625 pp., ISBN 978-0-8176-4948-7 A mathematical introduction to compressive sensing by Simon Foucart and Holger Rauhut [FR13] is about sparse solutions to systems of random linear equations. To begin, let me describe some striking phenomena that take place in this context. Afterward, I shall try to explain why these facts have captivated so many researchers over the last decade. I shall conclude with some comments on the book.

Compressed sensing with a jackknife and a bootstrap

“…To reconstruct an image from measurements taken in the Fourier domain (with independent and identically distributed centered complex Gaussian noise of standard deviation 0.02 √ 2 added to mimic machine imprecision), we minimize the sum of deviations from the measurements plus a total-variation regularizer via Algorithm 1 at the end of Section 2.2 of Tao and Yang (2009) (which is based on the work of Yang and Zhang ( 2011)), with 100 iterations, using the typical parameter settings µ = 10 12 and β = 10 (µ governs the fidelity to the measurements taken in the Fourier domain, and β is the strength of the coupling in the operator splitting for the alternating-direction method of multipliers). As discussed by Tropp (2017), this is the canonical setting for compressed sensing. All computations take place in IEEE standard double-precision arithmetic.…”

“…The requirement that gradients be concentrated on sparse subsets is sufficient but may not be necessary, and much recent research -including that of Hammernik et al (2018) -aims to generalize beyond this requirement by applying machine learning to representative data sets. Indeed, the literature on compressed sensing is vast and growing rapidly; see, for example, the recent review of Tropp (2017) for explication of all this and more.…”

Derandomizing Compressed Sensing With Combinatorial Design

“…Finally, one may argue that compressed sensing has not fully lived up to the high expectations of the community yet, see e.g. [12]. Arguably, one of the most glaring problems for applications is the requirement of choosing individual arXiv:1812.08130v1 [cs.IT] 19 Dec 2018 measurements at random 1 .…”

Compressed sensing with a jackknife and a bootstrap

“…To reconstruct an image from measurements taken in the Fourier domain (with independent and identically distributed centered complex Gaussian noise of standard deviation 0.02 added to mimic machine imprecision), we minimize the sum of deviations from the measurements plus a total-variation regularizer via Algorithm 1 at the end of Section 2.2 of [8] (which is based on the work of [11]), with 100 iterations, using the typical parameter settings µ = 10 12 and β = 10 (µ governs the fidelity to the measurements taken in the Fourier domain, and β is the strength of the coupling in the operator splitting for the alternating-direction method of multipliers). As discussed by [9], this is the canonical setting for compressed sensing. All computations take place in IEEE standard double-precision arithmetic.…”

“…The requirement that gradients be concentrated on sparse subsets is sufficient but may not be necessary, and much recent research aims to generalize beyond this requirement by applying machine learning to representative data sets. Indeed, the literature on compressed sensing is vast and growing rapidly; see, for example, the recent review of [9] for explication of all this and more.…”