Overview

'High-Dimensional Probability,' winner of the 2019 PROSE Award in Mathematics, offers an accessible and friendly introduction to key probabilistic methods for mathematical data scientists. Streamlined and updated, this second edition integrates theory, core tools, and modern applications. Concentration inequalities are central, including classical results like Hoeffding's and Chernoff's inequalities, and modern ones like the matrix Bernstein inequality. The book also develops methods based on stochastic processes – Slepian's, Sudakov's, and Dudley's inequalities, generic chaining, and VC-based bounds. Applications include covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, and machine learning. New to this edition are 200 additional exercises, alongside extra hints to assist with self-study. Material on analysis, probability, and linear algebra has been reworked and expanded to help bridge the gap from a typical undergraduate background to a second course in probability.

Full Product Details

9781009490641

Table of Contents

Reviews

'This book is a must-read for anyone interested in high-dimensional probability or its applications to data science. The second edition retains the enlightened selection of topics of the first edition, but with a more streamlined and self-contained exposition. The addition of an introductory chapter that serves as a refresher of basic concepts, and a wealth of new exercises at different levels should make the book appealing to an even broader audience.' Kavita Ramanan, Brown University 'With this second edition of his book on high-dimensional probability, Roman Vershynin has produced the reference book on the topic. Advanced students and practitioners interested in the mathematical foundations of data science will still enjoy the lively and progressive exposition of concepts of the first edition, with its worked examples and exercises, enriched by new introductory chapters and a collection of new enlightening exercises.' Rémi Gribonval, Inria & ENS de Lyon, France 'High-dimensional probability is a fascinating area of mathematics that unites probability and high-dimensional geometry—two beautiful, yet often counterintuitive, fields. It lies at the foundation of modern statistics, artificial intelligence, and machine learning. In this book, which has already become a classic, Roman Vershynin—both a leading researcher and a master expositor—presents the essential tools along with some of the central results and applications of high-dimensional probability. This work serves as an excellent textbook for graduate courses, sure to be appreciated by students in mathematics, statistics, computer science, and engineering. It is also an invaluable reference for researchers working in high-dimensional probability and statistics.' Elchanan Mossel, Massachusetts Institute of Technology 'The second edition of this excellent book is substantially enriched. It is a vital source of knowledge not only in probability but also in high-dimensional statistics.' Alexandre Tsybakov, CREST-ENSAE Paris 'Vershynin's High Dimensional Probability is a rare gem that transforms the rigorous landscape of high dimensional probability into an exciting and enjoyable journey. A must read for graduate students and researchers alike!' Van Vu, Yale University

Author Information

Roman Vershynin is Professor of Mathematics at the University of California, Irvine. He is an expert on randomness in mathematics and data science, especially in high-dimensional probability, statistics, and machine learning. His influential work has earned numerous honors including an invited ICM lecture, the Bessel Research Award, the IMS Medallion Award, and the 2019 PROSE Award for the first edition of this book.

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