A note on the Hanson-Wright inequality for random vectors with dependencies

Radoslaw Adamczak

doi:10.1214/ecp.v20-3829

Abstract

We prove that quadratic forms in isotropic random vectors $X$ in $\mathbb{R}^n$, possessing the convex concentration property with constant $K$, satisfy the Hanson-Wright inequality with constant $CK$, where $C$ is an absolute constant, thus eliminating the logarithmic (in the dimension) factors in a recent estimate by Vu and Wang. We also show that the concentration inequality for all Lipschitz functions implies a uniform version of the Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the inequalities by Borell, Arcones-Giné and Ledoux-Talagrand). Previous results of this type relied on stronger isoperimetric properties of $X$ and in some cases provided an upper bound on the deviations rather than a concentration inequality.In the last part of the paper we show that the uniform version of the Hanson-Wright inequality for Gaussian vectors can be used to recover a recent concentration inequality for empirical estimators of the covariance operator of $B$-valued Gaussian variables due to Koltchinskii and Lounici.

Highlights

The Hanson-Wright inequality asserts that if X1, . . . , Xn are independent mean zero, variance one random variables with sub-Gaussian tail decay, i.e. such that for all t > 0, P(|Xi| ≥ t) ≤ 2 exp(−t2/K2), and A = [aij]ni,j=1 is an n × n matrix, the quadratic form n Z=aij XiXj i,j=1 satisfies the inequalityP(|Z − trA| ≥ t) ≤ 2 exp − min t2 CK4 A C t K2 AKey words and phrases
For Gaussian variables it follows from estimates for general Banach space valued polynomials by Borell [8] and Arcones-Gine [4]
Since in many applications one considers quadratic forms in random vectors with dependencies among coefficients, some recent work has been devoted to proving counterparts of the Hanson-Wright inequality in a dependent setting

Summary

Introduction

Since in many applications one considers quadratic forms in random vectors with dependencies among coefficients, some recent work has been devoted to proving counterparts of the Hanson-Wright inequality in a dependent setting. One of the objectives of this paper is to remove the dependence on dimension in the above estimate (Theorem 2.3 below) as well as to prove corresponding uniform estimates for suprema of quadratic forms under some stronger assumptions on the random vector X (Theorem 2.4) Such uniform versions (corresponding to Banach space valued quadratic forms) for Gaussian random vectors were considered e.g. by Borell [8] and Arcones-Gine [4], whereas the Rademacher case was studied by Talagrand [30] and Bousquet-BoucheronLugosi-Massart [9].

Main results

Proofs of the main results

CK2 min