Existence of a single sparsity‑adaptive Hanson–Wright inequality for Bernoulli vectors across all regimes

Determine whether there exists a Hanson–Wright inequality for quadratic forms x^T A x of independent centered Bernoulli random variables with parameters p_i that is optimal uniformly across all sparsity regimes. Specifically, identify a single bound that adapts to p_i, recovers the classical Hanson–Wright behavior when p_i are constants, and remains sharp in the sparse regime p_i → 0.

Background

The literature offers several bounds for Bernoulli quadratic forms (e.g., Giné–Latała–Zinn (2000) and Schudy–Sviridenko (2011)), but they depend on different matrix norms and, when p_i are constant, do not recover the standard sub‑gaussian Hanson–Wright inequality.

The authors highlight the lack of a unified inequality that adapts to the sparsity level and is optimal in both dense and sparse regimes, motivating the question of whether such a single, regime‑adaptive inequality exists.