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High-Dimensional Probability

An Introduction with Applications in Data Science
BuchGebunden
296 Seiten
Englisch
Cambridge University Presserschienen am27.09.2018
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.mehr
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Produkt

KlappentextHigh-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.
ZusammenfassungAn integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
Details
ISBN/GTIN978-1-108-41519-4
ProduktartBuch
EinbandartGebunden
Erscheinungsjahr2018
Erscheinungsdatum27.09.2018
Seiten296 Seiten
SpracheEnglisch
Gewicht710 g
Artikel-Nr.47293592
Rubriken
GenreWirtschaft

Inhalt/Kritik

Inhaltsverzeichnis
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.mehr
Kritik
'This is an excellent and very timely text, presenting the modern tools of high-dimensional geometry and probability in a very accessible and applications-oriented manner, with plenty of informative exercises. The book is infused with the author's insights and intuition in this field, and has extensive references to the latest developments in the area. This book will be an extremely useful resource both for newcomers to this subject and for expert researchers.' Terence Tao, University of California, Los Angelesmehr

Autor

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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Vershynin, Roman