多项式阈值函数敏感性的小矩
Small moments of the sensitivity of polynomial threshold functions
AI总结：
本文研究多项式阈值函数的布尔表面积，通过递归方法和多项式限制引理，推导出敏感函数小矩的多对数上界，并观察到η=1/2处的相变。
摘要：
在Chang、Slote、Volberg和Zhang的论文\cite{BSA_of_PTF}的第一版中，作者修改了Kane在\cite{Correct_exponent_for_AS}中用于有界多项式阈值函数平均敏感性的优美递归方法。在\cite{BSA_of_PTF}中，Kane的论证被用于估计多项式阈值函数的布尔表面积。其桥梁是一个考虑所有平衡划分的组合平均引理。该引理替代了平均敏感性的可加性质。利用该引理，可以应用Kane型算法推导递推关系。求解递推关系后，得到布尔表面积的上界$e^{C_d \sqrt{\log n}}$。\n在同一论文的第二版中，作者推导了多项式阈值函数布尔表面积的多对数上界。区别在于他们使用了敏感函数的尾部估计。借助\cite{poly_restriction}中的多项式限制引理，他们改进了上界。值得注意的是，在应用多项式限制时，每个坐标独立且等概率地分配到每个部分。因此，划分不一定具有相等大小的块。换句话说，它可能不是平衡的。\n在本注记中，我们首先研究不同划分的效果。其次，我们使用第一版中的递归方法推导$\mathbb E[s(x)^η]$的多对数上界，其中$η< 1/2$。有趣的是，在两个版本的证明中（但以完全不同的形式），在$η=1/2$处发生了相变。第\ref{PhaseTr-s}节对此进行了处理。
英文摘要：
In the first version of Chang, Slote, Volberg, and Zhang's paper \cite{BSA_of_PTF}, the authors modify a nice recursive approach due to Kane in \cite{Correct_exponent_for_AS} where he bounded the average sensitivity of polynomial threshold functions. In \cite{BSA_of_PTF} Kane's argument was adopted to estimate the boolean surface area of polynomial threshold function. The bridge is a combinatorial averaging lemma considering all balanced partitions. The lemma serves as a substitute for an additive property of average sensitivity. With the lemma, one can apply a Kane-type algorithm to derive a recurrence. Solving the recurrence then gives an upper bound of $e^{C_d \sqrt{\log n}}$ for the boolean surface area. In the second version of the same paper, the authors derive a polylog upper bound for BSA of PTFs. The difference is that they use a tail estimate for the sensitivity function. With the help of a polynomial restriction lemma in \cite{poly_restriction} they sharpen the upper bound. It is noteworthy that when applying the polynomial restriction, each coordinate is put into each part independently with equal probability. As a result, a partition does not necessarily have equal-size blocks. In other words, it may not be balanced. In this note, we first investigate the effect of different partitioning. Second, we use the recursive method in the first version to derive a polylog upper bound for $\mathbb E[s(x)^η]$ where $η< 1/2$. It is interesting to note the phase transition that happens at $η=1/2$ in both versions of the proof (but in a completely different form). Section 2.5 treats that.
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