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,... 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.

翻译：我们重新审视并略微修改了高斯二次混沌浓度的 Hanson-Wright 不等式（HW 不等式）证明，从而能够通过将其表述中的绝对常数从当前已知的最大值 0.125 增加到至少 0.145（对称情形）来收紧界。我们还提出了所谓的 Laurent-Massart 不等式（LM 不等式）的一个更尖锐版本，通过该版本我们能够将 HW 不等式中的绝对常数从当前由 LM 不等式本身提供的最大值 0.134 增加到至少 0.152（半正定情形）。通过推广对称情形下的 HW 不等式，我们推导出由 m = 1, 2, 3,... 索引的高斯二次混沌的一系列浓度界，这些界涉及底层矩阵的 Schatten 范数。m = 1 的情形退化为 HW 不等式。这些界表现出行为上的相变，其意义在于：当偏差小于一个临界阈值时，m = 1 给出最紧的界；而当偏差大于上述阈值时，随着索引 m 的增加，界会变得越来越紧。最后，我们推导出一个在大小偏差区域都渐近地比 HW 不等式更紧的浓度界。