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High-dimensional probability: an introduction with applications in data science

By: Vershynin, Roman.
Material type: materialTypeLabelBookSeries: Cambridge series in statistical and probabilistic mathematics. Publisher: New Delhi Cambridge University Press 2018Description: xiv, 284 p. Includes bibliographical references and index.ISBN: 9781108415194.Subject(s): Probabilities | Stochastic processes | Random variablesDDC classification: 519.2 Summary: 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. https://www.cambridge.org/core/books/highdimensional-probability/797C466DA29743D2C8213493BD2D2102#fndtn-information
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Table of Contents:
Foreword
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 Theorem
Hints for Exercises
References
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Index
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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.

https://www.cambridge.org/core/books/highdimensional-probability/797C466DA29743D2C8213493BD2D2102#fndtn-information

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