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.

翻译：我们研究高维数据的统计韧性。针对固定（高维）矩阵M与共形Rademacher向量ξ构成的二次Rademacher混沌$\boldsymbol{\xi}^{\mathsf{T}} M \boldsymbol{\xi}$的反集中性质，本文给出对抗性噪声影响的估计。具体而言，我们探究ξ在多大程度上能承受对抗性符号翻转而不"膨胀"$\sup_{x\in \mathbb{R}} \mathbb{P} \left\{\boldsymbol{\xi}^{\mathsf{T}} M \boldsymbol{\xi} = x\right\}$，从而避免原始分布"去平滑化"、导致分布呈现更"颗粒化"且带对抗性偏倚的结果。本文给出了二次与双线性Rademacher混沌统计韧性的下界估计，并证明这些估计在关键参数区间内是渐近紧致的。