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.

翻译：(PTF) $f:\ mathbb{R}n\ rightrow \ mathb{R}$ 是表格 $f(x) =\ mathsf{sign} (p(x)) $是多度的多度值。 PTF是一个传统和研究周密的复杂类别,其应用范围包括复杂理论、学习理论、近似理论、量子复杂性等。我们解决了在 Goussian 空间设计多度阈值函数(PG) 的问题: 设计一个PRG, 以几位随机值和输出的种子为单位, 美元值为单位。 PTFTF是一个经典和研究周密的复杂类别, 其应用范围包括: $* O(1) log (n/\ varepslon) (PRG) (PG) 用于多位值值值值值值值值值值值值值值值值值的模拟生成器 。