A composition theorem for the Fourier Entropy-Influence conjecture

TL;DR

This paper proves a composition theorem for the Fourier Entropy-Influence conjecture, enabling the extension of its validity to new classes of Boolean functions and providing improved bounds on the ratio of spectral entropy to influence.

Contribution

It introduces a composition theorem for the FEI conjecture, allowing it to hold for complex functions built from simpler ones, and improves the known lower bound on the entropy-influence ratio.

Findings

FEI conjecture holds for read-once formulas over arbitrary gates of bounded arity.

Established a composition theorem for the FEI conjecture.

Constructed an explicit function with a ratio C ≥ 6.278, the largest known so far.

Abstract

The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96] seeks to relate two fundamental measures of Boolean function complexity: it states that $H[f] \leq C Inf[f]$ holds for every Boolean function $f$, where $H[f]$ denotes the spectral entropy of $f$, $Inf[f]$ is its total influence, and $C > 0$ is a universal constant. Despite significant interest in the conjecture it has only been shown to hold for a few classes of Boolean functions. Our main result is a composition theorem for the FEI conjecture. We show that if $g_1,...,g_k$ are functions over disjoint sets of variables satisfying the conjecture, and if the Fourier transform of $F$ taken with respect to the product distribution with biases $E[g_1],...,E[g_k]$ satisfies the conjecture, then their composition $F(g_1(x^1),...,g_k(x^k))$ satisfies the conjecture. As an application we show that the FEI conjecture holds for read-once formulas over arbitrary gates of bounded arity, extending a recent result [OWZ11] which proved it for read-once decision trees. Our techniques also yield an explicit function with the largest known ratio of $C \geq 6.278$ between $H[f]$ and $Inf[f]$, improving on the previous lower bound of 4.615.