多项式阈值函数灵敏度的界
Bounding the Sensitivity of Polynomial Threshold Functions
摘要：
给出了多项式阈值函数的平均灵敏度和噪声灵敏度的第一个非平凡上界。更具体地说，对于一个布尔函数f的n个变量等于一个总次数为d的实多元多项式的符号，我们证明了1）f的平均灵敏度至多为O（n ^{1 - 1/（4d +6）}）（我们还给出了一个组合证明的界限O（n ^{1 - 1/2^d}）。2)f对噪声率δ的噪声灵敏度至多为O（δ ^{1/（4d +6）}）。以前，只有线性情况下的界限是已知的。一路上，我们展示了新的结构定理的多项式阈值函数通过超压缩性获得的随机限制。这些结构的结果可能是独立的利益，因为它们提供了一个通用的模板转换相关的问题，多项式阈值函数定义的布尔超立方体的多项式阈值函数定义在高斯空间。
英文摘要：
We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial of total degree d we prove 1) The average sensitivity of f is at most O(n^{1-1/(4d+6)}) (we also give a combinatorial proof of the bound O(n^{1-1/2^d}). 2) The noise sensitivity of f with noise rate δis at most O(δ^{1/(4d+6)}). Previously, only bounds for the linear case were known. Along the way we show new structural theorems about random restrictions of polynomial threshold functions obtained via hypercontractivity. These structural results may be of independent interest as they provide a generic template for transforming problems related to polynomial threshold functions defined on the Boolean hypercube to polynomial threshold functions defined in Gaussian space.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
