High-Dimensional Probability (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 47)

A great read

This is definitely the best math book I read this year. It is at the same time a very well-written textbook as well as a great reference in the area. Excellent choice of topics, a joy to read, and especially valuable were the exercises throughout the book which makes it perfect for self-study since you can solve the exercises to internalize the ideas.

Great book! Personally I think this one much better than HDS. Hope to see the 2nd edition.

Vershynin's book covers a set of topics that is likely to become central in the education for "modern" mathematicians, statisticians, physicists, and (electrical) engineers. He discusses ideas, techniques, and tools that arise across fields, and he conceptually unifies them under the brand name of "high-dimensional probability". His choice of topics (e.g., concentration/deviation inequalities, random vectors/matrices, stochastic processes, etc.) and applications (e.g., sparse recovery, dimension-reduction, covariance estimation, optimization bounds, etc.) delivers a necessary (and timely) addition to the growing body of data-science-related literature—more on this below. Vershynin writes in a conversational, reader-friendly manner. He weaves theorems, lemmas, corollaries, and proofs into his dialogue with the reader without getting caught in an endless theorem-proof loop. In addition, the book's integrated exercises and its prompts to "check!" or think about "why?" are strong components of the book. My copy of the book is already full of notes to myself where I’m “checking” something or explaining “why” something is true/false. (Also, as an aside, I love that coffee cups are used to signal the difficulty of a problem—good style.) I want to highlight a few examples where Vershynin’s choice of topics and his prose shine brightly. In section 4.4.1, he guides us through an example that clearly illustrates the usefulness of ε-nets for bounding matrix norms. I’d seen ε-nets and covering numbers before, but never had good intuition for why they showed up in a proof. Similarly, I’d struggled to gain intuition about why/how Gaussian widths and Vapnik–Chervonekis dimension capture/measure the complexity of a set. After reading sections 7.5 and 8.3 and working through some exercises, the two concepts are much clearer. Moreover, Vershynin connects these ideas back to covering numbers, which helped me better my understanding of all three concepts. Finally, I found the discussions on chaining and generic chaining in chapter 8 to be excellent. Following them up with Talagrand’s comparison inequality, which becomes the hammer of choice for the matrix deviation inequality (in chapter 9), rounds out a long, but very valuable/useful chapter—and one that I’ll certainly re-study and reference. I would recommend this book for those interested in (high-dimensional) statistics, randomized numerical linear algebra, and electrical engineering (particularly, signal processing). As I'm coming to realize, the "concentration of measure" and “deviation inequality” toolbox is essential to these areas. Lastly, I believe that this book makes a great companion to “Concentration Inequalities” by Boucheron, Lugosi, Massart.

Thus far, I understand the book is the best one of making sense of the fundamentals of high dimensional probability, particularly of help to beginners.

An *excellent* first treatment of concepts in high-dimensional probability and statistics. The book is very clear and clean, and the many exercises (with helpful hints) make it a good resource for self-study. This has become one of my favorite math textbooks ever!

Great weekend reading!

The quality of the paper is bad. There are some photos of the paper quality, you can see. If you have ever used a gel pen in a typical A4, it's really similar to the quality the pages have.

O livro é muito bem escrito e com uma abordagem bem moderna.

The books that I end up buying in paper format are few in number, but they stay with me for years and are a regular source of information when I have to look up notes taken long ago. I am happy to welcome this book to my collection. The material is impeccable, presented in an engaging manner and sprinkled with occasional exercises. Working as a data scientist in industry, it provides a solid theoretical foundation for some of the algorithms I use (though do not expect a lot of directly applicable material). Some were criticizing the print quality. It may be there are different versions in circulation, but my copy is fine and the paper is reasonable thick and not seethrough under normal lighting conditions. Only criticism I have is that there are no solutions provided to the exercises. They are also not separately purchasable to my knowledge. This hampers it's usability as a self-study guide. For some reason, very few math books chose to also include solutions, I find this unfortunate. Would have been a clear five star book otherwise.

Excellent book, although there are a few typos here and there.