Sparse Hanson-Wright inequalities for subgaussian quadratic forms

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 $X=(X_1,\dots ,X_m)\in {\mathbb{R}}^m$ 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 ${\xi }_1,\dots ,{\xi }_m$ 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 $Y=HX$ where $H\in {\mathbb{R}}^{m\times m}$ is an $m \times m$symmetricmatrix;westudythelargedeviationboundonthe$\ell_2$-normof$D_{\xi} Y$fromitsexpectedvalue,whereforagivenvector$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.