Refining Concentration for Gaussian Quadratic Chaos with Applications in Sonar and Communications

Abstract

The paper studies concentration of measure for Gaussian quadratic chaos in the non-asymptotic regime where existing bounds are improved and new bounds are proposed. We begin by slightly tightening Hanson-Wright inequality (HWI) by increasing its absolute constant from the largest known value of 0.125 to at least 0.145 in the symmetric case. A sharper version of an inequality of Laurent and Massart (LMI) is presented. It results in an increase in the absolute constant in HWI from the largest available value of $1-\frac{\sqrt{3}}{2}$ due to LMI to $\frac{9-\sqrt{17}}{32}$ in the positive-semidefinite case. Moving beyond HWI, we develop a sequence of inequalities indexed by $m\ge1$ that involves Schatten norms of the underlying symmetric matrix. The case $m=1$ recovers HWI and the case $m=\infty$ leads to a novel bound called the $m_\infty$-bound. Avoiding Markov's inequality, we introduce the strong $\chi^2$-inequality and its loosened version, the weak $\chi^2$-inequality. To investigate the $m_\infty$-bound, we explore all concentration bounds that only involve the operator norm of the underlying positive-definite matrix. Five candidates are examined, namely, the $m_\infty$-bound, relaxed versions of HWI and LMI, the weak $\chi^2$-bound and the large deviations bound. The sharpest among these bounds is either the $m_\infty$-bound or the weak $\chi^2$-bound. If the matrix dimension is $n=2,4,6$, the weak $\chi^2$-bound is tighter than the $m_\infty$-bound. For even $n\ge8$, the $m_\infty$-bound is sharper than the weak $\chi^2$-bound if and only if the ratio of the tail parameter over the operator norm lies inside an open interval which expands indefinitely as $n$ grows. Modified versions of HW, $m_\infty$ and strong $\chi^2$ inequalities of various orders are proposed. Their effectiveness is demonstrated by two applications in signal detection for sonar and wireless communications.