次数- $d$多项式的高斯表面积和噪声敏感性
The Gaussian Surface Area and Noise Sensitivity of Degree-$d$ Polynomials
摘要：
我们提供了渐近尖锐的界限的高斯表面积和高斯噪声灵敏度的多项式阈值函数。特别是，我们表明，如果$f$是一个度$d$多项式阈值函数，那么它的高斯灵敏度在噪声率$ε$是小于一些数量渐近$\frac{d\sqrt{2ε}}π$和高斯表面积至多是$\frac{d}{\sqrt{2π}}$。此外，这些界限是渐进紧的，因为$ε\to 0$和$f$是$d$不同齐次线性函数乘积的阈值函数。
英文摘要：
We provide asymptotically sharp bounds for the Gaussian surface area and the Gaussian noise sensitivity of polynomial threshold functions. In particular we show that if $f$ is a degree-$d$ polynomial threshold function, then its Gaussian sensitivity at noise rate $ε$ is less than some quantity asymptotic to $\frac{d\sqrt{2ε}}π$ and the Gaussian surface area is at most $\frac{d}{\sqrt{2π}}$. Furthermore these bounds are asymptotically tight as $ε\to 0$ and $f$ the threshold function of a product of $d$ distinct homogeneous linear functions.
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