The Gaussian Surface Area and Noise Sensitivity of Degree-$d$ Polynomials

TL;DR

This paper establishes sharp bounds on the Gaussian surface area and noise sensitivity of degree-d polynomial threshold functions, revealing their asymptotic behavior and tightness for specific polynomial classes.

Contribution

It provides the first asymptotically tight bounds for Gaussian surface area and noise sensitivity of degree-d polynomial threshold functions.

Findings

Gaussian sensitivity at noise rate ε is asymptotically bounded by (d√(2ε))/π

Gaussian surface area is at most d/√(2π)

Bounds are tight for threshold functions of products of d distinct linear functions

Abstract

We provide asymptotically sharp bounds for the Gaussian surface area and the Gaussian noise sensitivity of polynomial threshold functions. In particular we show that if $f$ is a degree-$d$ polynomial threshold function, then its Gaussian sensitivity at noise rate $\epsilon$ is less than some quantity asymptotic to $\frac{d\sqrt{2\epsilon}}{\pi}$ and the Gaussian surface area is at most $\frac{d}{\sqrt{2\pi}}$. Furthermore these bounds are asymptotically tight as $\epsilon\to 0$ and $f$ the threshold function of a product of $d$ distinct homogeneous linear functions.