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.
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...