A Mathematical Introduction to Compressive Sensing (Applied by Simon Foucart

By Simon Foucart

At the intersection of arithmetic, engineering, and computing device technology sits the thriving box of compressive sensing. in line with the idea that facts acquisition and compression could be played concurrently, compressive sensing unearths purposes in imaging, sign processing, and lots of different domain names. within the components of utilized arithmetic, electric engineering, and theoretical laptop technology, an explosion of study job has already the theoretical effects that highlighted the potency of the elemental ideas. The dependent rules at the back of those rules also are of autonomous curiosity to natural mathematicians.

A Mathematical creation to Compressive Sensing offers an in depth account of the center idea upon which the sector is construct. With merely reasonable necessities, it's a great textbook for graduate classes in arithmetic, engineering, and computing device technological know-how. It additionally serves as a competent source for practitioners and researchers in those disciplines who are looking to collect a cautious knowing of the topic. A Mathematical creation to Compressive Sensing makes use of a mathematical point of view to give the middle of the speculation underlying compressive sensing.

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We refer to [39, 195, 271, 272, 294] for further information on the classical aspects. The use of sparse recovery techniques in sampling problems appeared early in the development of the compressive sensing theory [94, 97, 408, 409, 411, 416]. In fact, the alternative name compressive sampling indicates that compressive sensing can be viewed as a part of sampling theory—although it draws from quite different mathematical tools than classical sampling theory itself. Sparse approximation. The theory of compressive sensing can also be viewed as a part of sparse approximation with roots in signal processing, harmonic analysis 36 1 An Invitation to Compressive Sensing [170], and numerical analysis [122].

Appendix A covers background material from linear algebra and matrix analysis, including vector and matrix norms, eigenvalues and singular values, and matrix functions. Basic concepts and results from convex analysis and convex optimization are presented in Appendix B. We also treat matrix convexity and present a proof of Lieb’s theorem on the concavity of the matrix function X → tr exp(H + ln X) on the set of positive definite matrices. Appendix C presents miscellaneous material including covering numbers, Fourier transforms, elementary estimates on binomial coefficients, the Gamma function and Stirling’s formula, smoothing of Lipschitz functions via convolution, distributional derivatives, and differential inequalities.

A closely related regression problem arises in machine learning. Given random pairs of samples (tj , yj )m j=1 , where tj is some input parameter vector and yj is a scalar output, one would like to predict the output y for a future input data t. 2 Applications, Motivations, and Extensions 21 y = f (t) + e, where e is random noise. The task is to learn the function f based on training samples (tj , yj ). Without further hypotheses on f , this is an impossible task. Therefore, we assume that f has a sparse expansion in a given dictionary of functions ψ1 , .

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