High-Dimensional Probability: An Introduction with Applications in Data Science

Cambridge Series in Statistical and Probabilistic Mathematics #47

by Roman Vershynin

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.

Genres: Mathematics, Textbooks

277 pages, Unknown Binding

Reviews

[ShawnLeeZX · Rating 4 out of 5]

Overall, the book has an excellent structure, and offers interface to understand the field by connecting to applications. It also has numerous exercises and examples to aid understanding. The lost one star came from my doubt during reading the Dudley Entropy inequality. The author does not seem to adequately describe the deeper geometrical insight behind the inequality.

[smoh cat · Rating 5 out of 5]

Excellent structure. Very easy to follow. Read some of it for homework but ended up reading a lot of it just because how elegant some of the proof is.

[Abbas Mehrabian · Rating 5 out of 5]

Read my review here: https://www.amazon.ca/review/R2ZBSFKM...

[Yi Li · Rating 5 out of 5]

This is absolutely an excellent textbook to introduce students into, and master the basics of, the field of high-dimensional probability.

Chapters 1-7 are most well-written and nicely structured. The later chapters, such as Chapter 9 and 11, are a bit short and less elaborated, giving an unfinished feeling. This is a bit disappointing, because the author is a geometric functional analyst and I was expecting more geometric functional analysis could be discussed in those later chapters.

The overall writing of the book is concise, without a lot of details. Thus, beginners are strongly encouraged to do all the exercises (especially the one-star and two-star ones) in order to have a better understanding of the content and the details. Three-star exercises may involve useful tricks and can generally be solved in a reasonable amount of time (with a certain degree of math maturity). There are also a few four-star exercises, such as Exercise 3.4.5, 8.5.6, 10.6.11, which are absolutely challenging for beginners.

[Matthew Zhang · Rating 4 out of 5]

Decent but some of the exercises can get repetitive.

[Lucille Nguyen · Rating 5 out of 5]

Readable, useful book with good geometric intuition about the topics discussed. Has a strong eye towards applications.