Sparse Hanson-Wright inequalities for subgaussian quadratic forms

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 $ξ=(ξ_1, \ldots, ξ_m) \in \{0, 1\}^m$ be independent Bernoulli random variables. We prove the large deviation bound for a sparse quadratic form of $(X \circ ξ)^T A (X \circ ξ)$, where $A \in \mathbb{R}^{m \times m}$ is an $m \times m$ matrix, and random vector $X \circ ξ$ denotes the Hadamard product of an isotropic subgaussian random vector $X \in \mathbb{R}^m$ and a random vector $ξ\in \{0, 1\}^m$ such that $(X \circ ξ)_{i} = X_{i} ξ_i$, where $ξ_1, \ldots,ξ_m$ 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 $D_ξ Y$ 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.