Has the theoretical explanation of Compressed Sensing been simplified since 2005/2006?

Question

Is the theoretical explanation of compressed sensing given in the Candes, Romberg, and Tao papers from 2005 / 2006 still the simplest and most clear explanation available?

Are these original papers still the best resource for learning the theory behind compressed sensing? Which paper or textbook is the best starting point for understanding compressed sensing theory?

This is an interesting question, but I do believe it is about opinions and not directly about mathematics. — Hans Engler, Jun 14 '16 at 23:12
@HansEngler, there is a lot of questions about the best literature or textbooks on a given subject. Maybe some tags need to be added? — Yuriy S, Jun 15 '16 at 08:40
@YuriyS I'm not sure what the best tags would be, feel free to add any if you'd like. — eternalGoldenBraid, Jun 17 '16 at 21:50

score 7 · Accepted Answer · edited Oct 15 '19 at 10:51

A little late to the question, but there are a couple of good resources I found which were extremely useful in getting a good understanding of the field. Aside from the seminal papers you listed, I encourage you to take a look at the texts below. Note that CS is vast field, and there are countless papers on this subject, so do tailor your study to whatever objective you have.

$\textbf{1)}$ Holger Rauhut's text, "Compressive Sensing and Structured Random Matrices".

This text is available online for free when you do a google search of it. The first 100 pages give a very good overview of the different approaches in compressive sensing. It discusses popular conditions for uniform recovery of all s-sparse vectors through the RIP property and also mentions some other conditions which guarantee $l_0-l_1$ minimization equivalence. In addition, the first 100 pages also go into detail on non-uniform recovery via random sampling, which is an integral part of compressive sensing.

$\textbf{2)}$ "The Restricted Isometry Property and Its Implications for Compressed Sensing" by Emmanuel Candes

This short paper gives a very quick overview of Candes/Tao's seminal paper on RIP in compressive sensing. I find this to be very concise if you do not want to delve into the original paper, which is much longer. I think this is a good read for anyone interested in the field because RIP conditions are not only integral to the field, but the proofs to these elegant theorems can be understood with elementary mathematical knowledge.

$\textbf{3)}$ "RIPless theory of Compressive Sensing" by Candes/Plan

This paper contains more recent results to compressive sensing via random sampling. The field itself has moved quite a lot from 2005-2006, and this paper contains an overarching theory for when s-sparse vectors can be recovered exactly from $l_1$-minimization.

Has the theoretical explanation of Compressed Sensing been simplified since 2005/2006?

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