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24 results

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High-Dimensional Probability
  • An Introduction with Applications in Data Science
  • 2nd edition
  • Roman Vershynin
  • Coming soon
  • Expected online publication date:
    February 2026
    Print publication:
    31 March 2026
  • 'High-Dimensional Probability,' winner of the 2019 PROSE Award in Mathematics, offers an accessible and friendly introduction to key probabilistic methods for mathematical data scientists. Streamlined and updated, this second edition integrates theory, core tools, and modern applications. Concentration inequalities are central, including classical results like Hoeffding's and Chernoff's inequalities, and modern ones like the matrix Bernstein inequality. The book also develops methods based on stochastic processes – Slepian's, Sudakov's, and Dudley's inequalities, generic chaining, and VC-based bounds. Applications include covariance estimation, clustering, networks, semidefinite programming, coding, dimension reduction, matrix completion, and machine learning. New to this edition are 200 additional exercises, alongside extra hints to assist with self-study. Material on analysis, probability, and linear algebra has been reworked and expanded to help bridge the gap from a typical undergraduate background to a second course in probability.

  • Hamiltonicity of sparse pseudorandom graphs

  • Part of
  • Asaf Ferber, Jie Han, Dingjia Mao, Roman Vershynin
  • Journal:
    Combinatorics, Probability and Computing / Volume 34 / Issue 4 / July 2025
    Published online by Cambridge University Press:
    25 March 2025, pp. 596-620
    • Article
      • You have access
      • Open access
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  • We show that every $(n,d,\lambda )$-graph contains a Hamilton cycle for sufficiently large $n$, assuming that $d\geq \log ^{6}n$ and $\lambda \leq cd$, where $c=\frac {1}{70000}$. This significantly improves a recent result of Glock, Correia, and Sudakov, who obtained a similar result for $d$ that grows polynomially with $n$. The proof is based on a new result regarding the second largest eigenvalue of the adjacency matrix of a subgraph induced by a random subset of vertices, combined with a recent result on connecting designated pairs of vertices by vertex-disjoint paths in $(n,d,\lambda )$-graphs. We believe that the former result is of independent interest and will have further applications.

High-Dimensional Probability
  • An Introduction with Applications in Data Science
  • Roman Vershynin
  • Published online:
    29 September 2018
    Print publication:
    27 September 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.

  • 1 - Preliminaries on Random Variables

  • Roman Vershynin, University of California, Irvine
  • Book:
    High-Dimensional Probability
    Published online:
    29 September 2018
    Print publication:
    27 September 2018, pp 5-10
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