Sparse Hanson–Wright inequalities for subgaussian quadratic forms

Abstract

In this paper, we provide a proof for the Hanson–Wright inequalities for sparse 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 = Xiξi, 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 = HX where H ∈ Rm×m is an m × m symmetric matrix; we study the large deviation bound on the 2-norm DξY22 from its expected value, where for a given vector x ∈ Rm, Dx = diag(x) denotes the diagonal matrix whose main diagonal entries are the entries of x. This form arises naturally from the context of covariance estimation.