This is a textbook in probability in high dimensions with a view toward applications in data sciences.
It is intended for doctoral and advanced masters students and beginning researchers
in mathematics, statistics, electrical engineering, computer science,
computational biology and related areas, who are looking to expand their knowledge of theoretical methods
used in modern research in data sciences.
Why this book?
Data sciences are moving fast, and probabilistic methods often provide a foundation and inspiration for such advances.
A typical graduate probability course is no longer sufficient to acquire the level of
mathematical sophistication that is expected from a beginning researcher in data sciences today.
The proposed book intends to partially cover this gap. It presents some of the key probabilistic methods and results
that should form an essential toolbox for a mathematical data scientist. This book can be used as a textbook
for a basic second course in probability with a view toward data science applications. It is also suitable
The essential prerequisites for reading this book are a rigorous course
in probability theory (on Masters or Ph.D. level), an excellent command of undergraduate linear algebra,
and general familiarity with basic notions about Hilbert and normed spaces and linear operators.
Knowledge of measure theory is not essential but would be helpful.
I am Professor of Mathematics at the University of Michigan
and an expert in theoretical and applied high-dimensional probability.
I study probabilistic structures that appear across mathematics
and data sciences, in particular random matrix theory, geometric functional
analysis, convex and discrete geometry, high-dimensional statistics,
information theory, learning theory, signal processing, numerical analysis,
and network science.
May 23, 2017. An "Appetizer" added to the front of the book. It presents the so-called Maurey's empirical method,
which is an elegant and elementary application of probability to bound covering numbers of sets.
Chapter 7 is now polished.
April 27, 2017. Chapter 6 is now polished.
April 20, 2017. Chapter 5 is now polished. I cleaned up the guarantees
of covariance estimation both in this chapter and those appeared earlier in Chapter 4.
February 23, 2017. Chapter 4 is now polished. I added an application to error correction
codes in Section 4.3 and rewrote the application for covariance estimation in Section 4.7.
February 9, 2017. Chapter 3 is now polished. I added a section (3.7) on
kernel methods and Krivine's proof of Grothendieck's inequality, which gives (almost) the best known
bound on the constant.
January 20, 2017. Chapter 2 is now polished.
January 4, 2017. Chapter 1 has been polished. The difficulty of exercises will be indicated by the number
of coffee cups one may need to solve them.
December 21, 2016. Numerous typos and inaccuracies fixed throughout the book.
It was then converted into the publisher's style, which miraculously reduced the number of pages by 50!
December 20, 2016. A short version of this book, condensed into just four lectures,
can be found here.
November 15, 2016. Two big sections are added in Chapter 8:
VC dimension and applications in statistical learning theory.
October 24, 2016. A few applications are added to Chapter 3:
Grothendieck's inequality, semidefinite programming, and maximum cut for graphs.