Improved Concentration Bounds for Gaussian Quadratic Forms

TL;DR

This paper introduces a new, tighter concentration inequality for Gaussian quadratic forms involving monotonic functions, enabling more accurate p-value bounds in high-dimensional data analysis.

Contribution

It develops a Chernoff-style concentration inequality for a broad class of Gaussian quadratic forms with computationally efficient trace-based bounds.

Findings

Bounds are significantly tighter than previous results

Numerical examples demonstrate improved accuracy

Applicable to high-dimensional screening tasks

Abstract

For a wide class of monotonic functions $f$, we develop a Chernoff-style concentration inequality for quadratic forms $Q_f \sim \sum\limits_{i=1}^n f(\eta_i) (Z_i + \delta_i)^2$, where $Z_i \sim N(0,1)$. The inequality is expressed in terms of traces that are rapid to compute, making it useful for bounding p-values in high-dimensional screening applications. The bounds we obtain are significantly tighter than those that have been previously developed, which we illustrate with numerical examples.