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on June 2, 2011
The book starts with a prologue of an under-determined linear system and how sparsity constraints help to solve it with the use of a Langrangian. Next the authors introduce the key idea of how certain norms promote sparsity. There are some good diagrams that really help the geometric intuition (though not as good as the ones by Donoho et al. in connection with Lasso). I really love the way they motivate and frame the entire field but still appeal to concept that most people who have studied linear algebra can relate to.

The first 6 chapters are a master piece in pedagogy. Except for the not so-standard usage of Spark as the measurement of coherence among elements of a dictionary. Mutual coherence is common and easier to grasp since it directly address the size of inner products. This leads to a rather jarring switch when RIP is introduced.

I am still puzzled why the authors do not appeal to frame theory. That leads to strange looking reference to self-dual frames and tight frames when the book never talked about frames.

I also wonder why the authors did not cite Boyd's great book. The treatment of log-barrier was sort of just another penalty function. The term log-barrier was never used in the book.

Overall I cannot put the book down and was especially grateful to the authors for introducing iterative shrinkage as a central theme to link many modern numerical algorithms to solve the basic sparse optimization problem.
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on June 20, 2013
This book is about sparse representations. Mathematically it is all about solving:

min ||x||_0 subject to Ax=b

where ||x||_0 is a special type of "norm", it counts the nonzero entries in a vector x. And the issue at hand is that only a few columns in A will (multiplied by x) result in b. In other words, in many practical circumstances - apparently - this vector x only requires a few nonzero entries.

I am only on a third or so of the book (after one weekend), so I've to adjust my review later. Until now the authors do focus on trying to get theoretical grips on this topic. When is there a sparse solution? If you have one, can you find an even sparser one. I find many angles very interesting. That an uncertainty principles leads to a uniqueness result is amazing. The authors subsequently introduce the concept of a "spark" and are able to say if for example matching pursuit will succeed in recovering the sparsest solution.

In general the authors are using math all over the place, so if you don't like math stay away from the book. However, they take a very gentle approach from my perspective (as a robotics engineer), making quite some intermediate steps explicit. Of course, I have to go back some pages so now and then, but it's worthwhile. And I look forward to the second part of the book that describes the image processing applications, which after skimming looks much less "math-heavy", but where I hope the authors maintained the same pleasant level of detail.
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on May 28, 2011
Covers the essentials of sparse approximation in a clear and practical style. The sections on basis and matching pursuit algorithms are especially good. There aren't many textbooks on this material and this (with Mallat's "A Wavelet Tour of Signal Processing") is very valuable to mathematicians/engineers/computer scientists working in compressed sensing.
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on May 8, 2013
I'm a Math and Physics PhD from UCSD and I find this too much math and too little signal processing. Might not be a fair criticism but I didn't need to see a lot of proofs that IMHO are too brief to easily follow and too long to be interesting. If I was going to "fix" this, I'd skip the proofs and show a lot of toy examples for the algorithms: you know a 2x 5 matrix with small integers run for a few iterations. As it is too much theory to be practical, too much algo for a math book and so on...
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on December 31, 2012
this book is very good. It contain some useful examples and provide software so as to implement most of methods in it.
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