A note on the improved sparse Hanson-Wright inequalities

Abstract: In this paper, we establish sparse Hanson-Wright inequalities for quadratic forms of sparse $\alpha$-sub-exponential random vectors where $\alpha \in (0,2]$. When only considering the regime $0 < \alpha \leq 1$, we derive a sharper sparse Hanson-Wright inequality that achieves optimality in certain special cases. These results generalize some classical Hanson-Wright inequalities without sparse structure.