Sparse Hanson-Wright Inequality for a Bilinear Form of Sub-Gaussian Variables

TL;DR

This paper introduces a new Hanson-Wright inequality tailored for sparse bilinear forms of sub-Gaussian variables, enabling improved statistical testing and error control in high-dimensional data with missing values.

Contribution

It generalizes existing deviation inequalities to sparse bilinear forms and applies these results to cross-covariance testing under missing data conditions.

Findings

Derived a new concentration inequality for sparse bilinear forms

Enabled threshold determination for correlation testing with error control

Extended analysis to multiplicative measurement error scenarios

Abstract

In this paper, we derive a new version of Hanson-Wright inequality for a sparse bilinear form of sub-Gaussian variables. Our results are generalization of previous deviation inequalities that consider either sparse quadratic forms or dense bilinear forms. We apply the new concentration inequality to testing the cross-covariance matrix when data are subject to missing. Using our results, we can find a threshold value of correlations that controls the family-wise error rate. Furthermore, we discuss the multiplicative measurement error case for the bilinear form with a boundedness condition.