A note on the improved sparse Hanson-Wright inequalities

Abstract

We establish sparse Hanson-Wright inequalities for quadratic forms of sparse $\hat{I}\pm$-sub-exponential random vectors with exponent parameter $\hat{I}\pm \in (0,2]$. In the regime $0<\hat{I}\pm \le 1$ we derive a refined inequality that is optimal in several canonical models. These results extend the classical Hanson-Wright bound to the sparse setting. Illustrative applications include covariance matrix estimation with incomplete observations, low-rank matrix approximation under the maximum norm with sparsified sketches, and concentration inequalities for sparse $\hat{I}\pm$-sub-exponential random vectors.