高斯空间中PTF的随机约束和PRG
Random restrictions and PRGs for PTFs in Gaussian Space
摘要：
多项式阈值函数（PTF）$f:\mathbb{R}^n \rightarrow \mathbb{R}$是形式为$f(x) = \mathsf{sign}(p(x))$的函数，其中$p$是次数至多为$d$的多项式。PTF是一个经典的、研究得很好的复杂性类，其应用涉及复杂性理论、学习理论、近似理论、量子复杂性等。我们解决了在高斯空间中为多项式阈值函数（PTF）设计伪随机发生器（PRG）的问题：设计一个PRG，它取几位随机数的种子，并输出一个$n$维向量，其分布与标准的多变量高斯的$d$ PTF度是不可区分的.我们的主要结果是一个PRG，它采取了一个种子的$d^{O(1)}\log ( n / \varepsilon)\log(1/\varepsilon)/\varepsilon^2$随机比特的输出，不能区分$n$维高斯分布的优势比$\varepsilon$度$d$ PTF。由于O'Donnell，Servedio和Tan（STOC'20）的最佳先前生成器具有准多项式依赖性（即，$d^{O(\log d)}$的种子长度）的程度为$d$。一路上，我们证明了几个近紧的PTF的限制，可能是独立的利益的结构特性。
英文摘要：
A polynomial threshold function (PTF) $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a function of the form $f(x) = \mathsf{sign}(p(x))$ where $p$ is a polynomial of degree at most $d$. PTFs are a classical and well-studied complexity class with applications across complexity theory, learning theory, approximation theory, quantum complexity and more. We address the question of designing pseudorandom generators (PRG) for polynomial threshold functions (PTFs) in the gaussian space: design a PRG that takes a seed of few bits of randomness and outputs a $n$-dimensional vector whose distribution is indistinguishable from a standard multivariate gaussian by a degree $d$ PTF. Our main result is a PRG that takes a seed of $d^{O(1)}\log ( n / \varepsilon)\log(1/\varepsilon)/\varepsilon^2$ random bits with output that cannot be distinguished from $n$-dimensional gaussian distribution with advantage better than $\varepsilon$ by degree $d$ PTFs. The best previous generator due to O'Donnell, Servedio, and Tan (STOC'20) had a quasi-polynomial dependence (i.e., seedlength of $d^{O(\log d)}$) in the degree $d$. Along the way we prove a few nearly-tight structural properties of restrictions of PTFs that may be of independent interest.
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