低权半空间的Fooling交
Fooling intersections of low-weight halfspaces
摘要：
一个权$t$半空间是一个布尔函数$f(x)=$符号$(w_1 x_1 + \cdots + w_n x_n - θ)$，其中每个$w_i$是一个整数在$\{-t,\dots,t\}.$我们给出了一个显式的伪随机生成器，$δ$ -愚弄任何交叉的$k$权$t$半空间与种子长度多$(\log n, \log k,t,1/δ)$ .特别是，我们的结果给出了一个显式的PRG，愚弄任何准多$(n)$数量的半空间的任何多$\log(n)$权重的任何$1/$多$\log(n)$精度使用种子长度聚$\log(n).$在这项工作之前，没有显式的PRG与非平凡的种子长度是已知的，即使是愚弄$n$权重-1半空间的交叉点，以恒定的精度。我们的PRG融合技术的分析，从两个不同的工作线上的无条件伪随机不同种类的布尔函数。本文推广了Harsha，Klivans和Meka \cite{HKM12}关于正则半空间的交的愚弄方法，并将此方法与Bazzi \cite{Bazzi:07}和Razborov \cite{Razborov:09}关于有界独立愚弄CNF公式的结果相结合.我们的分析引入新的耦合为基础的成分到标准的Lindeberg方法建立定量的中心极限定理和相关的伪随机结果。
英文摘要：
A weight-$t$ halfspace is a Boolean function $f(x)=$sign$(w_1 x_1 + \cdots + w_n x_n - θ)$ where each $w_i$ is an integer in $\{-t,\dots,t\}.$ We give an explicit pseudorandom generator that $δ$-fools any intersection of $k$ weight-$t$ halfspaces with seed length poly$(\log n, \log k,t,1/δ)$. In particular, our result gives an explicit PRG that fools any intersection of any quasipoly$(n)$ number of halfspaces of any poly$\log(n)$ weight to any $1/$poly$\log(n)$ accuracy using seed length poly$\log(n).$ Prior to this work no explicit PRG with non-trivial seed length was known even for fooling intersections of $n$ weight-1 halfspaces to constant accuracy. The analysis of our PRG fuses techniques from two different lines of work on unconditional pseudorandomness for different kinds of Boolean functions. We extend the approach of Harsha, Klivans and Meka \cite{HKM12} for fooling intersections of regular halfspaces, and combine this approach with results of Bazzi \cite{Bazzi:07} and Razborov \cite{Razborov:09} on bounded independence fooling CNF formulas. Our analysis introduces new coupling-based ingredients into the standard Lindeberg method for establishing quantitative central limit theorems and associated pseudorandomness results.
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