A Noisy-Influence Regularity Lemma for Boolean Functions

We present a regularity lemma for Boolean functions f : {−1, 1}n → {−1, 1} based on noisy influence, a measure of how locally correlated f is with each input bit. We provide an application of the regularity lemma to weaken the conditions on the Majority is Stablest Theorem. We also prove a “homogenized” version stating that there is a set of input bits so that most restrictions of f on those bits have small noisy influences. These results were sketched out by [OSTW10], but never published. With their permission, we present the full details here.