Sparse Hanson-Wright inequalities for subgaussian quadratic forms

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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 = (X₁,, Xₘ) ∈ Rᵐ be a random vector with independent subgaussian components, and ξ=(ξ₁,, ξₘ) ∈ {0, 1}ᵐ be independent Bernoulli random variables. We prove the large deviation bound for a sparse quadratic form of (X ∘ ξ)ᵗ A (X ∘ ξ), where A ∈ Rᵐ × ᵐ is an m × m matrix, and random vector X ∘ ξ denotes the Hadamard product of an isotropic subgaussian random vector X ∈ Rᵐ and a random vector ξ∈ {0, 1}ᵐ such that (X ∘ ξ)ᵢ = Xᵢ ξᵢ, where ξ₁,,ξₘ 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 ∈ Rᵐ× ᵐ is an m × m symmetric matrix; we study the large deviation bound on the ℓ₂-norm of Dξ Y from its expected value, where for a given vector x ∈ Rᵐ, Dₓ denotes the diagonal matrix whose main diagonal entries are the entries of x. This form arises naturally from the context of covariance estimation.