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

TL;DR

This paper refines the Hanson-Wright inequality for dependent isotropic random vectors, removing dimension-dependent factors, and extends it to a uniform version applicable to quadratic forms and empirical covariance estimators.

Contribution

It proves a dimension-free Hanson-Wright inequality for dependent vectors with convex concentration and extends it to uniform bounds for quadratic forms and covariance estimators.

Findings

Eliminates logarithmic dimension factors in Hanson-Wright inequality for dependent vectors.

Establishes a uniform Hanson-Wright inequality for quadratic forms based on Lipschitz concentration.

Recovers recent covariance estimation bounds for Gaussian vectors using the uniform Hanson-Wright inequality.

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-Gin\'e 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.