Fooling Gaussian PTFs via Local Hyperconcentration

TL;DR

This paper introduces a new pseudorandom generator that effectively fools degree-$d$ polynomial threshold functions over Gaussian space with a seed length polynomial in $d$ and logarithmic in $n$, improving over previous exponential dependencies.

Contribution

The paper presents a novel Local Hyperconcentration Theorem and a pseudorandom generator with significantly reduced seed length for Gaussian polynomial threshold functions.

Findings

Seed length is polynomial in $d$ and logarithmic in $n$

Achieves fooling of degree-$d$ Gaussian PTFs with improved seed efficiency

Introduces the Local Hyperconcentration Theorem for Gaussian polynomials

Abstract

We give a pseudorandom generator that fools degree-$d$ polynomial threshold functions over $n$-dimensional Gaussian space with seed length $\mathrm{poly}(d)\cdot \log n$. All previous generators had a seed length with at least a $2^d$ dependence on $d$. The key new ingredient is a Local Hyperconcentration Theorem, which shows that every degree-$d$ Gaussian polynomial is hyperconcentrated almost everywhere at scale $d^{-O(1)}$.