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.
Closes the gap between the standard probability curriculum and what mathematical data scientists need to know
Selects the core ideas and methods and presents them systematically with modern motivating applications to bring readers quickly up to speed
Features integrated exercises that invite readers to sharpen their skills and build practical intuition
Preface
Appetizer: using probability to cover a geometric set
1. Preliminaries on random variables
2. Concentration of sums of independent random variables
3. Random vectors in high dimensions
4. Random matrices
5. Concentration without independence
6. Quadratic forms, symmetrization and contraction
7. Random processes
8. Chaining
9. Deviations of random matrices and geometric consequences
10. Sparse recovery
11. Dvoretzky-Milman's theorem
Bibliography
Index.
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