High-Dimensional Probability

'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.

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.