We study the statistical resilience of high-dimensional data. Our results
provide estimates as to the effects of adversarial noise over the
anti-concentration properties of the quadratic Radamecher chaos
$\boldsymbol{\xi}^{\mathsf{T}} M \boldsymbol{\xi}$, where $M$ is a fixed
(high-dimensional) matrix and $\boldsymbol{\xi}$ is a conformal Rademacher
vector. Specifically, we pursue the question of how many adversarial sign-flips
can $\boldsymbol{\xi}$ sustain without "inflating" $\sup_{x\in \mathbb{R}}
\mathbb{P} \left\{\boldsymbol{\xi}^{\mathsf{T}} M \boldsymbol{\xi} = x\right\}$
and thus "de-smooth" the original distribution resulting in a more "grainy" and
adversarially biased distribution. Our results provide lower bound estimations
for the statistical resilience of the quadratic and bilinear Rademacher chaos;
these are shown to be asymptotically tight across key regimes.

我们研究高维数据的统计弹性，结果提供敌对噪声对二次 Radamecher 混沌的反集中性质的影响估计量，其中 M 是一个固定的（高维）矩阵，ξ 是一个符合射线混合向量。具体而言，我们研究了 ξ 可以承受多少敌对符号翻转而不会 “膨胀”σ 为上证∈RP {ξTMξ=x}，从而使原始分布变得更 “粒状” 和带有对这个分布的敌对偏差。我们的结果提供了二次和双线性 Rademacher 混乱的统计弹性下界估计，这些下界在关键范围内被证明是渐近紧密的。