Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems

By Michel Talagrand. 2014

242 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7060411d507a5439ef837f257ceea1a341449f8d

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Citations in the corpus

Nonparametric Shape-Restricted Regression (2017, background): “More sophisticated generic chaining arguments exist which gives tight bounds (up to universal multiplicative constants) for suprema of Gaussian processes; see [130].”
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression (2018, methodology): “…norm and the generic chaining proof techniques in Section 10.2 of Talagrand (2014) and Section 5 of Dirksen (2015), one can obtain “optimal” tail bounds on the supremum of the empirical processes under a sub-Weibull envelope assumption in terms of the γ-functionals of Talagrand (2014).”
Sharp nonasymptotic bounds on the norm of random matrices with independent entries (2014, background): “This idea mirrors the basic ingredients in the general theory of Bernoulli processes [22], Chapter 5.”
Bridging Theory and Algorithm for Domain Adaptation (2019, methodology): “To this end, we describe the notion of covering number from Zhou (2002); Anthony & Bartlett (2009); Talagrand (2014).”
Convergence rates of least squares regression estimators with heavy-tailed errors (2017, methodology): “The bounds in [45] use techniques from generic chaining [58], and work particularly well for “sub-Gaussian classes” (defined in [45]).”
Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method (2018, unknown): “By Theorem 2.2.26 in Talagrand (2014) it holds for t ≥ 0 and an absolute constant C > 0 that with probability at least 1 − C exp(−t) sup A∈A,γ∈Sn−1 γTAX . diam(ASn−1, ρ) √ t+ γ2(ASn−1, ρ), where diam(ASn−1) = sup x,y∈ASn−1 ‖x−y‖max i ‖Xi‖ψ2 ≤ 2 sup A∈A ‖A‖maxi ∥∥Xi ∥∥ ψ2 and the functional γ2 is…”
Chaining Mutual Information and Tightening Generalization Bounds (2018, methodology): “149]) and later developed by Dudley, Fernique, Talagrand and many others [11], the basic idea of chaining is to first describe the dependencies between the hypotheses by a metric d on the set T , then to discretize T and to approximate the maximal value (maxt∈T ht(G)) by approximating the maxima ove”
Sub-Gaussian estimators of the mean of a random matrix with heavy-tailed entries (2016, background): “We conclude this section by recalling the notion of Talagrand’s generic chaining complexity (see [Tal14]) and several related results.”
Minimax Statistical Learning and Domain Adaptation with Wasserstein Distances (2017, background): “Hence, X is subgaussian with respect to dΦ, and therefore the Rademacher average Rn(Φ) can be upper-bounded by the Dudley entropy integral [12]: Rn(Φ) ≤ 12 √ n ∫ ∞”