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High-Dimensional Probability

An Introduction with Applications in Data Science

Who is this book for?

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 for self-study.


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.

Roman Vershynin

Roman Vershynin

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.

Have a look at my University webpage to learn more about me.

Where Can You Get the Book?

Once this book is completed, it is going to be published by Cambridge University Press. If you want to be notified once the book is available, please send me an e-mail.

Download Draft

I am still writing this book. The current draft of the textbook is available:

(Warning: large file, please be patient with download.)

As of now, the technical material is almost complete. References and a lot of "chat" will be added; please also ignore the typos in the current version.

This draft is updated periodically. Use it at your own risk, and only for your personal and classroom needs. Please do not distribute the copy.


Got ideas how this textbook can be improved? Want to suggest useful topics or exercises? Please let me know, I will be happy to hear what you think.