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Cambridge Series in Statistical and Probabilistic Mathematics: Series Number 47: High-Dimensional Pr
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Cambridge Series in Statistical and Probabilistic Mathematics: Series Number 47: High-Dimensional Pr

Cambridge Series in Statistical and Probabilistic Mathematics: Series Number 47: High-Dimensional Pr

Roman Vershynin

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296 pages, parution le 14/11/2018
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Indisponible

Résumé

The data sciences are moving fast, and probabilistic methods are both the foundation and a driver. This highly motivated text brings beginners up to speed quickly and provides working data scientists with powerful new tools. Ideal for a basic second course in probability with a view to data science applications, it is also suitable for self-study.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.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.Roman Vershynin is Professor of Mathematics at the University of California, Irvine. He studies random geometric structures across mathematics and data sciences, in particular in random matrix theory, geometric functional analysis, convex and discrete geometry, geometric combinatorics, high-dimensional statistics, information theory, machine learning, signal processing, and numerical analysis. His honors include an Alfred Sloan Research Fellowship in 2005, an invited talk at the International Congress of Mathematicians in Hyderabad in 2010, and a Bessel Research Award from the Humboldt Foundation in 2013. His 'Introduction to the Non-Asymptotic Analysis of Random Matrices' has become a popular educational resource for many new researchers in probability and data science.

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Éditeur(s) Cambridge University Press
Auteur(s) Roman Vershynin
Parution 14/11/2018
Nb. de pages 296
EAN13 9781108415194

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