Fooling Gaussian PTFs via local hyperconcentration

Abstract

We give a pseudorandom generator that fools degree-d polynomial threshold functions over n-dimensional Gaussian space with seed length dO(logd) · logn. All previous generators had a seed length with at least a 2d dependence on d. The key new ingredient is our Local Hyperconcentration Theorem, which shows that every degree-d Gaussian polynomial is hyperconcentrated almost everywhere at scale d-O(logd).