Fooling Gaussian PTFs via Local Hyperconcentration

03/13/2021

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by   Ryan O'Donnell, et al.

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We give a pseudorandom generator that fools degree-d polynomial threshold functions over n-dimensional Gaussian space with seed length poly(d)·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).