A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions
Penghui Yao, Mingnan Zhao
TL;DR
This work establishes an explicit pseudorandom generator that $\varepsilon$-fools any function of $k$ degree-$d$ polynomial threshold functions over Gaussian space with seed length $\mathrm{poly}(k,d,1/\varepsilon)\cdot\log n$. The construction decomposes into two main components: (i) bounded-independence Gaussian vectors $Y_i$ whose average $Y$ fools the target class, with $R$-wise independence and $R=O(\log(kd/\varepsilon))$, and (ii) a discretization step that replaces $Y$ by a finite-entropy, $O(dR)$-wise independent approximation $X$ while preserving fooling guarantees. The analysis combines a mollifier-based reduction, a hybrid argument across steps, Taylor expansions in Gaussian space, Hermite expansions, and anti-concentration hypercontractivity to control error terms. The result yields an explicit PRG with seed length polynomial in $k$, $d$, and $1/\varepsilon$ times $\log n$, enabling derandomization for a broad family of low-degree PTFs on Gaussian space and offering a route to efficient constructions via discretization of Gaussian sources.
Abstract
Developing explicit pseudorandom generators (PRGs) for prominent categories of Boolean functions is a key focus in computational complexity theory. In this paper, we investigate the PRGs against the functions of degree-$d$ polynomial threshold functions (PTFs) over Gaussian space. Our main result is an explicit construction of PRG with seed length $\mathrm{poly}(k,d,1/ε)\cdot\log n$ that can fool any function of $k$ degree-$d$ PTFs with probability at least $1-\varepsilon$. More specifically, we show that the summation of $L$ independent $R$-moment-matching Gaussian vectors $ε$-fools functions of $k$ degree-$d$ PTFs, where $L=\mathrm{poly}( k, d, \frac{1}ε)$ and $R = O({\log \frac{kd}ε})$. The PRG is then obtained by applying an appropriate discretization to Gaussian vectors with bounded independence.