Positive spectrahedra: Invariance principles and Pseudorandom generators

TL;DR

This paper develops explicit pseudorandom generators for positive spectrahedra, extending previous work on polytopes, by establishing an invariance principle, noise sensitivity bounds, and applications to learning and discrepancy problems.

Contribution

It introduces a novel invariance principle for positive spectrahedra using the Lindeberg method, a first in this context, and constructs efficient PRGs with polylogarithmic seed length.

Findings

Constructed PRGs with seed length polylogarithmic in parameters.

Proved an invariance principle for positive spectrahedra.

Established bounds on noise sensitivity and a Littlewood-Offord theorem.

Abstract

In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit pseudorandom generators (PRGs) for arbitrary $m$-facet polytopes in $n$ variables with seed length poly-logarithmic in $m,n$, concluding a sequence of works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, Zuckerman (SICOMP 2013) for fooling linear and polynomial threshold functions, respectively. In this work, we consider a natural extension of PRGs for intersections of positive spectrahedrons. A positive spectrahedron is a Boolean function $f(x)=[x_1A^1+\cdots +x_nA^n \preceq B]$ where the $A^i$s are $k\times k$ positive semidefinite matrices. We construct explicit PRGs that $\delta$-fool "regular" width-$M$ positive spectrahedrons (i.e., when none of the $A^i$s are dominant) over the Boolean space with seed length $\textsf{poly}(\log k,\log n, M, 1/\delta)$. Our main technical contributions are the following: We first prove an invariance principle for positive spectrahedra via the well-known Lindeberg method. As far as we are aware such a generalization of the Lindeberg method was unknown. Second, we prove an upper bound on noise sensitivity and a Littlewood-Offord theorem for positive spectrahedra. Using these results, we give applications for constructing PRGs for positive spectrahedra, learning theory, discrepancy sets for positive spectrahedra (over the Boolean cube) and PRGs for intersections of structured polynomial threshold functions.