High-Dimensional Probability

By Roman Vershynin. 2018

1,076 citations recorded by the source corpus.

'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.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8b7f6f60af3c369d068b86f55a61692b3305d078

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Citations in the corpus

Minimax optimality of permutation tests (2020, background): “This quadratic form has been well-studied in the literature through a decoupling argument (e.g. Chapter 6 of Vershynin, 2018), which leads to the following theorem.”
Information Regularized Neural Networks (2018, background): “6, Vershynin [2018]], let M,m denote upper and lower bounds of the integrand of (27) correspondingly, we have”
Confounding-Robust Policy Improvement (2018, methodology): “…of Lemma 3 with a contraction argument, and obtain a bound on the maxima of the absolute value deviation by an argument of Remark 8.1.5 of Vershynin (2018): note that the zero function is an element of the class of non-decreasing functions on R, and apply Dudley’s inequality to the…”
Sub‐Weibull distributions: Generalizing sub‐Gaussian and sub‐Exponential properties to heavier tailed distributions (2019, positive): “Our characterizations are inspired by the characterizations of sub-Gaussian and sub-Exponential distributions in Vershynin (2018).”
Distance geometry and data science (2019, neutral): “2015; Vershynin 2018).”
Annealed Denoising Score Matching: Learning Energy-Based Models in High-Dimensional Spaces (2019, background): “…the learning process in relation to the data manifold X , we recall two elementary properties of random Gaussian vectors in high-dimensional spaces: First, the length distribution of random vectors becomes concentrated at √ dσ (Vershynin, 2018), where σ2 is the variance of a single dimension.”
Exact minimax risk for linear least squares, and the lower tail of sample covariance matrices (2019, background): “We have, denoting where the ﬁrst inequality follows from a comparison of covering and packing numbers [Ver18, Lemma 4.2.8], the second one from the inclusion S d 1 ⊂ B d and the last one from a volumetric argument [Ver18, Lemma 4.2.13].”
Randomized numerical linear algebra: Foundations and algorithms (2020, background): “R. Van Handel (2016), Probability in high dimension, Apc 550 lecture notes, Princeton Univ. R. Vershynin (2018), High-dimensional probability, Vol. 47 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge.”
Securing Distributed Gradient Descent in High Dimensional Statistical Learning (2018, background): “Thirdly, certain distributional assumptions are inevitable to some extent, in order to show the learning procedures perform well on the average case beyond the worst case guarantees [Ver18].”