Improve the sparse Hanson–Wright bound in the sub‑gaussian case

Determine whether the tail bound in Theorem 2 (the sparse α‑subexponential Hanson–Wright inequality) can be strengthened when the multipliers ξ_i are sub‑gaussian (α = 2). Concretely, for independent random variables X_i = x_i ξ_i with x_i ∼ Ber(p_i) and centered sub‑gaussian ξ_i, ascertain whether one can improve the third regime in the bound for P(|X^T A X| ≥ t) beyond the current factor exp(−c (t/(K^2 ||A||_max))^{1/2}), while retaining the variance‑sensitive and sparsity‑adaptive first two terms.

Background

Theorem 2 extends Hanson–Wright inequalities to random vectors with sparse α‑subexponential entries, yielding a three‑regime tail bound. The third term behaves like exp(−c (t/(K^2 ||A||_max))^{min{α/2, 1/2}}). For α > 1, and in particular for the sub‑gaussian case α = 2, this exponent is 1/2 and does not improve with α.

The authors note that in the sub‑gaussian sparse setting, Zhou (2019) obtained a bound with a second‑regime term t/(K^2 ||A||) independent of sparsity but without the 1/2‑exponent regime. The open question asks whether the present Theorem 2 can be sharpened specifically for sub‑gaussian multipliers, potentially improving the third regime while preserving the variance‑sensitive and sparsity‑aware structure of the other terms.