Polylogarithmic-dependence PRGs for thresholds of halfspaces

Establish whether an explicit pseudorandom generator can achieve polylogarithmic dependence on the number of halfspaces for the broader class of thresholds of halfspaces, improving on the superlinear dependence supplied by prior constructions.

Background

The paper studies explicit pseudorandom generators for functions formed by applying a threshold operation to multiple halfspaces, focusing particularly on $k$-out-of-$m$ thresholds of halfspaces. Prior work by Gopalan, O'Donnell, Wu, and Zuckerman provides a generator for monotone functions of halfspaces, but its seed length has superlinear dependence on the number $m$ of halfspaces.

Generators for intersections of halfspaces achieve seed lengths with polylogarithmic dependence on $m$, but the analyses of those constructions do not directly extend to thresholds of halfspaces. The paper addresses this gap by showing that the O'Donnell–Servedio–Tan generator for polytopes also fools $k$-out-of-$m$ thresholds, with a seed length that is polylogarithmic in $m$ when $k$ is polylogarithmic in $m$. The broader question of obtaining polylogarithmic dependence on $m$ for this class was explicitly left unresolved by prior work.