On the Analysis of Boolean Functions and Fourier-Entropy-Influence Conjecture

TL;DR

This paper reviews classical results and presents new bounds on the entropy of Boolean functions, specifically relating entropy to influence measures, advancing understanding of the Fourier-Entropy-Influence conjecture.

Contribution

It introduces a novel upper bound on the entropy of Boolean functions in terms of influence and influence-specific terms, contributing to the analysis of the Fourier-Entropy-Influence conjecture.

Findings

Entropy of Boolean functions is bounded by influence measures.

New upper bound involving sum over influence-specific logarithmic terms.

Provides insights into the Fourier-Entropy-Influence conjecture.

Abstract

This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that readers could probably read it independently. Among the new results, what is the most remarkable is that we prove that the entropy of a boolean function $f$ could be upper bounded by $O(I(f))+O(\sum\limits_{k}I_k(f)\log (1/I_k(f)))$.