The Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian Processes

By R. Dudley. 1967

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Citations in the corpus

Additive models with trend filtering (2017, positive): “Then on E , Rn ( F ∩B2(t) ) ≤ Eσ [ sup g∈F∩Bn( √ 2t) 1 n ∣∣∣∣∣ n∑ i=1 σig(zi) ∣∣∣∣∣ ] ≤ c√ n ∫ √2t 0 √ logN(δ, ‖ · ‖n,F) dδ ≤ c √ Kρw/2√ n ∫ √2t 0 δ−w/2 dδ = c√ n t1−w/2, where in second line we used Dudley’s entropy integral [Dudley, 1967], and in the third line we used Assumption C3.”
On the Error of Random Fourier Features (2015, methodology): “In order to apply the method of Dudley (1967), we must work around (cid:107) ω i (which appears in the covariance of the error process) being potentially unbounded.”
Sliced Score Matching: A Scalable Approach to Density and Score Estimation (2019, neutral): “Then, Dudley’s entropy integral (Dudley, 1967) gives”
Efficient Policy Learning (2017, positive): “Given this assumption define the complexity κ of the class Π in terms of a variant of the classical entropy integral of Dudley (1967):6 κ (Π) = ∫ 1 0 √ log (NH(ε2, Π)) dε.”
Adaptive computation and machine learning (1998, background): “This problem can be avoided either by assuming the existence of an envelope, that is a single non-negative function with a finite expectation lying above the absolute value of the loss of every function in the hypothesis set [Dudley, 1984, Pollard, 1984, Dudley, 1987, Pollard, 1989, Haussler, 1992],”
Long Range Dependence (2007, methodology): “The power-like behaviour of the incremental variance immediately allows one to check the metric entropy condition (see Dudley (1967)) or the Kolmogorov criterion (see e.”
Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation (2019, methodology): “…functions in F i ’s: Textbook results (Bartlett & Mendelson, 2002) bound the generalization error by the Rademacher complexity (formally deﬁned in Section A) of the family of losses L , which in turn is bounded by the covering number of L through Dudley’s entropy integral theorem (Dudley, 1967).”
Confidence sets in sparse regression (2012, unknown): “By Dudley’s bound [see Dudley (1967) or more recent references such as van der Vaart and Wellner (1996), van de Geer (2000)], applied to the (conditional on X) Gaussian process θ ′ → εT X(θ − θ ′), and using ∫ c0 √log(c/x) dx = c ∫ 1 0 √ log(1/x) dx = cA, where A is the constant A = ∫ 10 √log(1/x)…”
Policy Learning With Observational Data (2017, unknown): “cy value obtained by learning with doubly robust scores over the class of depth-2 trees. C Proofs C.1 Proof of Lemma2 Our proof of this result follows the outline of the classical chaining argument ofDudley (1967), whereby we construct a sequence of approximating sets of increasing precision for Ae ”
Penalized Estimation in Additive Regression with High-Dimensional Data (2017, background): “|hε,f1in|/C1 >n −1/2ψ n(δ,F1)+δ p t/n ) ≤ exp(−t), where C1 = C1(D0,D1) >0 is a constant, depending only on (D0,D1). S2.2 Dudley and Talagrand inequalities The following inequalities are due to Dudley (1967) and Talagrand (1996). Lemma 13. For δ>0, let F1 be a functional class such that sup”
The Convex Geometry of Linear Inverse Problems (2010, methodology): “Another tool for computing Gaussian widths is based on Dudley’s inequality [32, 49], which bounds the width of a set in terms of the covering number of the set at all scales.”
Rademacher complexity and spin glasses: A link between the replica and statistical theories of learning (2019, background): “Indeed it can be shown ([11, 12, 13]) that the Rademacher complexity can be bounded by the VC dimension so that for some constant value C , Rm(F) ≤ C √ dVC(F) m .”
Some new asymptotic theory for least squares series: Pointwise and uniform results (2012, unknown): “(η 1 = 1) = P(η 1 = −1) = 1/2) that are independent of the data, and denote η= (η 1,...,ηn). Also, let Eη[·] denote the expectation with respect to the distribution of η. Then by Dudley’s inequality (Dudley, 1967), Eη sup x∈X |α(x)′[Qb−1 −I]Gn[ηipiǫi]| . Zθ 0 q logN(T,k ·kn,2,ε)dε, where θ:= 2sup ”