Bounding the Sensitivity of Polynomial Threshold Functions

TL;DR

This paper establishes the first non-trivial bounds on the average and noise sensitivity of polynomial threshold functions, extending understanding beyond linear cases and introducing structural theorems via hypercontractivity.

Contribution

It provides the first bounds for polynomial threshold functions of degree d, and introduces structural theorems connecting Boolean and Gaussian polynomial threshold functions.

Findings

Average sensitivity is at most O(n^{1-1/(4d+6)})

Noise sensitivity at rate δ is at most O(δ^{1/(4d+6)})

Structural theorems relate Boolean and Gaussian polynomial threshold functions

Abstract

We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial of total degree d we prove 1) The average sensitivity of f is at most O(n^{1-1/(4d+6)}) (we also give a combinatorial proof of the bound O(n^{1-1/2^d}). 2) The noise sensitivity of f with noise rate \delta is at most O(\delta^{1/(4d+6)}). Previously, only bounds for the linear case were known. Along the way we show new structural theorems about random restrictions of polynomial threshold functions obtained via hypercontractivity. These structural results may be of independent interest as they provide a generic template for transforming problems related to polynomial threshold functions defined on the Boolean hypercube to polynomial threshold functions defined in Gaussian space.