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

## TL;DR

This paper develops uniform Hanson-Wright concentration inequalities for unbounded subgaussian entries using the entropy method, leading to new bounds and applications in covariance estimation, matrix Bernstein inequalities, and Ising models.

## Contribution

It introduces a novel entropy method approach for uniform Hanson-Wright inequalities applicable to unbounded entries, extending existing results and enabling new applications.

## Key findings

- Established a tighter Hanson-Wright inequality in certain regimes.
- Derived a dimension-free matrix Bernstein inequality for subexponential spectral norm matrices.
- Applied inequalities to covariance estimation with missing data and Ising models.

## 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 coordinates of $X$ itself. Our results recover, in particular, the classic uniform bound of Talagrand (1996) for Rademacher chaoses and the more recent uniform result of Adamczak (2015), 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 techniques 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 (2014). 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 (2003).