High-Dimensional Probability: An Introduction with Applications in Data Science (Cambridge Series in Statistical and Probabilistic Mathematics Book 47) by Roman Vershynin | (PDF) Free Download

Description

High-dimensional probability offers insight into the behavior of random vectors, random matrices, random subspaces, and objects used to quantify uncertainty in high dimensions. Drawing on ideas from probability, analysis, and geometry, it lends itself to applications in mathematics, statistics, theoretical computer science, signal processing, optimization, and more. It is the first to integrate theory, key tools, and modern applications of high-dimensional probability. Concentration inequalities form the core, and it covers both classical results such as Hoeffding’s and Chernoff’s inequalities and modern developments such as the matrix Bernstein’s inequality. It then introduces the powerful methods based on stochastic processes, including such tools as Slepian’s, Sudakov’s, and Dudley’s inequalities, as well as generic chaining and bounds based on VC dimension. A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, machine learning, compressed sensing, and sparse regression.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book is very easy to follow up and the content is advancedHighly recommendVershynin’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.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.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!As has been covered in the other reviews, this is a really excellent text and I almost feel guilty breaking the string of five-star reviews. That being said, something nobody has commented on is the print quality of the book. In regular lighting conditions, you can clearly see the text on the other side of the page. I found this distracting enough to switch to reading a pdf copy from the university library. I thought I may have gotten a knock-off copy, but I ordered it directly from Amazon, not a third-party seller. It’s a little puzzling because I also have Wainwright’s High-Dimensional Statistics from the same publisher, and it does not have this issue. I’ve attached a couple of pictures so you can judge for yourself before purchasing.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.This is a very well-written book for people who are interested in understanding the geometric aspects of modern data science. Personally I would say I am very much influenced by this book as well as many other papers by the author. He is a great and inspiring mathematician.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.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.O livro é muito bem escrito e com uma abordagem bem moderna.

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