Sparse Hanson-Wright inequalities for subgaussian quadratic forms

TL;DR

This paper establishes new concentration inequalities for sparse quadratic forms of subgaussian vectors, extending Hanson-Wright inequalities to settings involving sparsification and sampling, with applications in covariance estimation.

Contribution

It provides the first proof of Hanson-Wright inequalities for sparsified subgaussian quadratic forms, addressing two types of sparsity and their large deviation bounds.

Findings

Proves large deviation bounds for sparsified quadratic forms of subgaussian vectors.

Establishes concentration inequalities for sampled anisotropic subgaussian vectors.

Extends Hanson-Wright inequalities to sparse and sampled subgaussian settings.

Abstract

In this paper, we provide a proof for the Hanson-Wright inequalities for sparsified quadratic forms in subgaussian random variables. This provides useful concentration inequalities for sparse subgaussian random vectors in two ways. Let $X = (X_1, \ldots, X_m) \in \mathbb{R}^m$ be a random vector with independent subgaussian components, and $\xi =(\xi_1, \ldots, \xi_m) \in \{0, 1\}^m$ be independent Bernoulli random variables. We prove the large deviation bound for a sparse quadratic form of $(X \circ \xi)^T A (X \circ \xi)$, where $A \in \mathbb{R}^{m \times m}$ is an $m \times m$ matrix, and random vector $X \circ \xi$ denotes the Hadamard product of an isotropic subgaussian random vector $X \in \mathbb{R}^m$ and a random vector $\xi \in \{0, 1\}^m$ such that $(X \circ \xi)_{i} = X_{i} \xi_i$, where $\xi_1, \ldots,\xi_m$ are independent Bernoulli random variables. The second type of sparsity in a quadratic form comes from the setting where we randomly sample the elements of an anisotropic subgaussian vector $Y = H X$ where $H \in \mathbb{R}^{m\times m}$ is an $m \times m$ symmetric matrix; we study the large deviation bound on the $\ell_2$-norm of $D_{\xi} Y$ from its expected value, where for a given vector $x \in \mathbb{R}^m$, $D_{x}$ denotes the diagonal matrix whose main diagonal entries are the entries of $x$. This form arises naturally from the context of covariance estimation.