Uniform Hanson-Wright type concentration inequalities for unbounded entries via the entropy method

Electronic Journal of Probability. 2020. Vol. 25. P. 1–30.

Zhivotovskiy N.

This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector X∈RnX∈Rn 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 XX 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 XX. 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].

Keywords: concentration inequalities, modified logarithmic Sobolev inequalities, uniform Hanson-Wright inequalities, Rademacher chaos

Publication based on the results of: Uncertainty quantification in high-dimensional models (2020)