Hanson–Wright inequality in Banach spaces

Adamczak, Radosław; Latała, Rafał; Meller, Rafał

doi:10.1214/19-aihp1041

Cited by 8 publications

(19 citation statements)

References 27 publications

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“…This is a version of the famous Hanson-Wright inequality. For the various forms of the Hanson-Wright inequality we refer to [2,4,30,32,47,55,57]. Note that by a modification of our proofs (using arguments especially adapted to polynomials), it is possible to replace |A abs | op by |A| op , thus avoiding the drawback of switching to a matrix with a possibly larger operator norm.…”

Section: Resultsmentioning

confidence: 99%

Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities

2020

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In this paper, we prove multilevel concentration inequalities for bounded functionals f = f (X 1 , . . . , X n ) of random variables X 1 , . . . , X n that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f . We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes f (X ) = sup g∈F |g(X )| and suprema of homogeneous chaos in bounded random variables in the Banach space caseThe latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for U -statistics with bounded kernels h and for the number of triangles in an exponential random graph model. Keywords Concentration of measure • Empirical processes • Functional inequalities • Hamming cube • Logarithmic Sobolev inequality • Product spaces • Suprema of chaos • Weakly dependent random variables Mathematics Subject Classification (2010) 60E15 • 05C80 This research was supported by the German Research Foundation (DFG) via CRC 1283 "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications".

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Section: Resultsmentioning

confidence: 99%

Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities

2020

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“…Unfortunately we are able to show (2.4) only for d = 2 and with an additional factor ln p (cf. [2]). It is likely that by a modification of our proof one can show (2.4) for arbitrary d with an additional factor (ln p) C(d) .…”

Section: Notation and Main Resultsmentioning

confidence: 99%

“…Therefore, it is natural to seek inequalities which are expressed in terms of deterministic quantities and expectations of some F -valued polynomial chaoses, but do not involve expectations of additional suprema of such polynomials. This was the motivation behind the article [2], concerning the case d = 2 and containing the following bound, valid for p ≥ 1…”

Section: Moments Of Gaussian Chaoses In Banach Spacessmentioning

confidence: 99%

“…Let us also remark that this approach cannot be used to pass from d = 1, i.e., the inequality (1.3), to d = 2, i.e., the inequality (1.6) (see Remark 3.7). While the proof of (1.6) presented in [2] relies on a similar set of tools (chaining arguments, Gaussian concentration) some of the technical estimates of entropy numbers are obtained differently than in the induction step…”

Section: Moments Of Gaussian Chaoses In Banach Spacessmentioning

confidence: 99%

“…In a recent paper [2] we considered Gaussian quadratic forms with coefficients in a Banach space and obtained upper bounds on their tails and moments, expressed in terms of quantities which are easier to deal with. In the real valued case our estimates reduce to the Hanson-Wright inequality, and for a large class of Banach-spaces (related to Pisier's Gaussian property α and containing all type 2 spaces) they may be reversed.…”

Section: Introductionmentioning

confidence: 99%

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