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2020 Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method
Yegor Klochkov, Nikita Zhivotovskiy
Electron. J. Probab. 25: 1-30 (2020). DOI: 10.1214/20-EJP422

Abstract

This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector $X \in \mathbb{R} ^{n}$ with independent subgaussian components. The core technique of the paper is based on the entropy method combined with truncations of both gradients of functions of interest and of the components of $X$ itself. Our results recover, in particular, the classic uniform bound of Talagrand [28] for Rademacher chaoses and the more recent uniform result of Adamczak [2] which holds under certain rather strong assumptions on the distribution of $X$. We provide several applications of our techniques: we establish a version of the standard Hanson-Wright inequality, which is tighter in some regimes. Extending our results we show a version of the dimension-free matrix Bernstein inequality that holds for random matrices with a subexponential spectral norm. We apply the derived inequality to the problem of covariance estimation with missing observations and prove an almost optimal high probability version of the recent result of Lounici [21]. Finally, we show a uniform Hanson-Wright-type inequality in the Ising model under Dobrushin’s condition. A closely related question was posed by Marton [22].

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Yegor Klochkov. Nikita Zhivotovskiy. "Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method." Electron. J. Probab. 25 1 - 30, 2020. https://doi.org/10.1214/20-EJP422

Information

Received: 13 December 2018; Accepted: 25 January 2020; Published: 2020
First available in Project Euclid: 8 February 2020

zbMATH: 1445.60019
MathSciNet: MR4073683
Digital Object Identifier: 10.1214/20-EJP422

Subjects:
Primary: 60F10

JOURNAL ARTICLE
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