Heat and Noise on Cubes and Spheres: The Sensitivity of Randomly Rotated Polynomial Threshold Functions

TL;DR

This paper explores the relationship between spherical harmonics and Fourier functions over hypercubes, providing bounds on noise sensitivity of polynomial threshold functions under random rotations, with implications for the Gotsman-Linial conjecture.

Contribution

It introduces a novel bound linking spherical sensitivity and Boolean noise sensitivity, and applies it to prove an average case of the Gotsman-Linial conjecture.

Findings

Bound on expected Boolean noise sensitivity in terms of spherical sensitivity

Establishment of a relationship between spherical heat equation and function sensitivity

Proof of an average case of the Gotsman-Linial conjecture

Abstract

We establish a precise relationship between spherical harmonics and Fourier basis functions over a hypercube randomly embedded in the sphere. In particular, we give a bound on the expected Boolean noise sensitivity of a randomly rotated function in terms of its "spherical sensitivity," which we define according to its evolution under the spherical heat equation. As an application, we prove an average case of the Gotsman-Linial conjecture, bounding the sensitivity of polynomial threshold functions subjected to a random rotation.