Improved Concentration Bounds for Gaussian Quadratic Forms

Abstract

For a wide class of monotonic functions $f$, we develop a Chernoff-style concentration inequality for quadratic forms $Q_f\sim \sum _{i=1}^{n}f({\eta }_i)(Z_i+{\delta }_i{)}^2$, 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.

12 pages, 1 figure