Random restrictions and PRGs for PTFs in Gaussian Space

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.