Refining Concentration for Gaussian Quadratic Chaos

Abstract: We visit and slightly modify the proof of Hanson-Wright inequality (HW inequality) for concentration of Gaussian quadratic chaos where we are able to tighten the bound by increasing the absolute constant in its formulation from its largest currently known value of 0.125 to at least 0.145 in the symmetric case. We also present a sharper version of the so-called Laurent-Massart inequality (LM inequality) through which we are able to increase the absolute constant in HW inequality from its largest currently available value of $0.134$ due to LM~inequality itself to at least $0.152$ in the positive-semidefinite case. Generalizing HW inequality in the symmetric case, we derive a sequence of concentration bounds for Gaussian quadratic chaos indexed over $m=1,2,3,\cdots$ that involves the Schatten norms of the underlying matrix. The case $m=1$ reduces to HW inequality. These bounds exhibit a phase transition in behaviour in the sense that $m=1$ results in the tightest bound if the deviation is smaller than a critical threshold and the bounds keep getting tighter as the index $m$ increases when the deviation is larger than the aforementioned threshold. Finally, we derive a concentration bound that is asymptotically tighter than HW inequality both in the small and large deviation regimes. Finally, we explore concentration bounds when the underlying matrix is positive-semidefinite and only the dimension~$n$ and its operator norm (largest eigenvalue) are known. Four candidates are examined, namely, the $m_\infty$-bound, relaxed versions of HW and LM bounds and a bound that we refer to as the $\chi^2$-bound.