Refining Concentration for Gaussian Quadratic Chaos

We slightly modify the proof of Hanson-Wright inequality (HWI) for concentration of Gaussian quadratic chaos where we tighten the bound by increasing the absolute constant in its formulation from the largest known value of 0.125 to at least 0.145 in the symmetric case. We also present a sharper version of an inequality due to Laurent and Massart (LMI) through which we increase the absolute constant in HWI from the largest available value of approximately $0.134$ due to LMI itself to at least $0.152$ in the positive-semidefinite case. A new sequence of concentration bounds indexed by $m=1,2,3,\cdots, \infty$ is developed that involves Schatten norms of the underlying matrix. The case $m=1$ recovers HWI. These bounds undergo a phase transition in the sense that if the tail parameter is smaller than a critical threshold $\tau_c$, then $m=1$ is the tightest and if it is larger than $\tau_c$, then $m=\infty$ is the tightest. This leads to a novel bound called the~$m_\infty$-bound. A separate concentration bound named twin to HWI is also developed that is tighter than HWI for both sufficiently small and large tail parameter. Finally, we explore concentration bounds when the underlying matrix is positive-semidefinite and only the dimension~$n$ and its largest eigenvalue are known. Five candidates are examined, namely, the $m_\infty$-bound, relaxed versions of HWI and LMI, the $\chi^2$-bound and the large deviations bound. The sharpest among these is always either the $m_\infty$-bound or the $\chi^2$-bound. The case of even dimension is given special attention. If $n=2,4,6$, the $\chi^2$-bound is tighter than the $m_\infty$-bound. If $n$ is an even integer greater than or equal to 8, the $m_\infty$-bound is sharper than the $\chi^2$-bound if and only if the ratio of the tail parameter over the largest eigenvalue lies inside a finite open interval which expands indefinitely as $n$ grows.