A note on the improved sparse Hanson-Wright inequalities

TL;DR

This paper extends the Hanson-Wright inequalities to sparse $ ext{alpha}$-sub-exponential random vectors, providing refined bounds that are optimal in certain models, with applications in covariance estimation and matrix approximation.

Contribution

It introduces sparse Hanson-Wright inequalities for $ ext{alpha}$-sub-exponential vectors, including a refined version for $0< ext{alpha} extless 1$, extending classical bounds to sparse settings.

Findings

Derived optimal refined inequalities for $0<\alpha\le 1$

Extended Hanson-Wright bounds to sparse $ ext{alpha}$-sub-exponential vectors

Applied results to covariance estimation and matrix approximation

Abstract

We establish sparse Hanson-Wright inequalities for quadratic forms of sparse $\alpha$-sub-exponential random vectors with exponent parameter $\alpha\in(0, 2]$. In the regime $0< \alpha\le 1$ 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 $\alpha$-sub-exponential random vectors.