A note on the improved sparse Hanson-Wright inequalities

TL;DR

The paper develops sparse Hanson–Wright inequalities for quadratic forms of sparse α-sub-exponential vectors with α ∈ (0,2], delivering a main bound for 0<α≤2 and a refined, tighter bound for 0<α≤1. The authors combine decoupling, contraction, and sparse Bernstein tools to handle both diagonal and off-diagonal contributions, yielding explicit tail decay in terms of weighted matrix norms and sparsity parameters. They demonstrate the utility of these bounds through three applications: covariance estimation with missing observations, sparsified-sketch-based low-rank approximation, and concentration results for sparse α-sub-exponential vectors, including RIP-type guarantees and improved tail behavior over prior sparse HW results. The work connects and extends prior results in sparse probability, providing optimality insights in certain regimes and highlighting practical implications for high-dimensional inference under sparsity and heavy tails.

Abstract

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

Theorems & Definitions (12)

• proof
• proof
• proof : Proof of Theorem \ref{['Theo_main1']}
• proof : Proof of Theorem \ref{['Theo_main2']}
• proof
• proof
• proof : Proof of Theorem \ref{['thm:RIP']}
• proof : Proof of Theorem \ref{['Theo_application_low']}
• proof
• proof : Proof of Theorem \ref{['Theorem_application']}