An Invariance Principle for Polytopes

TL;DR

This paper establishes a new invariance principle for polytopes with a polylogarithmic dependence on the number of faces, leading to improved bounds on noise sensitivity and the construction of efficient pseudorandom generators.

Contribution

It introduces a novel invariance principle for polytopes with a polylogarithmic dependence on the number of faces, improving upon previous linear bounds.

Findings

Polylogarithmic bound on the difference in probabilities for polytopes under different distributions.

Construction of pseudorandom generators that fool polytopes with polylogarithmic seed length.

First deterministic quasi-polynomial algorithms for counting solutions to certain integer programs.

Abstract

Let X be randomly chosen from {-1,1}^n, and let Y be randomly chosen from the standard spherical Gaussian on R^n. For any (possibly unbounded) polytope P formed by the intersection of k halfspaces, we prove that |Pr [X belongs to P] - Pr [Y belongs to P]| < log^{8/5}k * Delta, where Delta is a parameter that is small for polytopes formed by the intersection of "regular" halfspaces (i.e., halfspaces with low influence). The novelty of our invariance principle is the polylogarithmic dependence on k. Previously, only bounds that were at least linear in k were known. We give two important applications of our main result: (1) A polylogarithmic in k bound on the Boolean noise sensitivity of intersections of k "regular" halfspaces (previous work gave bounds linear in k). (2) A pseudorandom generator (PRG) with seed length O((log n)*poly(log k,1/delta)) that delta-fools all polytopes with k faces with respect to the Gaussian distribution. We also obtain PRGs with similar parameters that fool polytopes formed by intersection of regular halfspaces over the hypercube. Using our PRG constructions, we obtain the first deterministic quasi-polynomial time algorithms for approximately counting the number of solutions to a broad class of integer programs, including dense covering problems and contingency tables.