Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- Appetizer Using Probability to Cover a Set
- 1 A Quick Refresher on Analysis and Probability
- 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 on Sets
- Hints for the Exercises
- References
- Index
Appetizer Using Probability to Cover a Set
Published online by Cambridge University Press: 30 January 2026
Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- Appetizer Using Probability to Cover a Set
- 1 A Quick Refresher on Analysis and Probability
- 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 on Sets
- Hints for the Exercises
- References
- Index
Summary
This chapter begins with Maurey’s empirical method – a probabilistic approach to constructing economical convex combinations. We apply it to bound covering numbers and the volumes of polytopes, revealing their counterintuitive behavior in high dimensions. The exercises refine these bounds and culminate in the Carl–Pajor theorem on the volume of polytopes.
Keywords
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- Chapter
- Information
- High-Dimensional ProbabilityAn Introduction with Applications in Data Science, pp. 1 - 7Publisher: Cambridge University PressPrint publication year: 2026
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