# Fourier Entropy-Influence Conjecture for Random Linear Threshold Functions

## TL;DR

This paper proves that the Fourier-Entropy-Influence Conjecture holds with high probability for random linear threshold functions under natural weight distributions, advancing understanding of the conjecture's scope.

## Contribution

It demonstrates that the FEI conjecture is valid for random linear threshold functions with weights drawn from uniform or normal distributions.

## Key findings

- FEI conjecture holds for random LTFs with high probability
- Validates FEI for weights from uniform distribution
- Validates FEI for weights from normal distribution

## Abstract

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function $f:\{+1,-1\}^n \to \{+1,-1\}$, the Fourier entropy of $f$ is at most its influence up to a universal constant factor. While the FEI conjecture has been proved for many classes of Boolean functions, it is still not known whether it holds for the class of Linear Threshold Functions. A natural question is: Does the FEI conjecture hold for a `random' linear threshold function? In this paper, we answer this question in the affirmative. We consider two natural distributions on the weights defining a linear threshold function, namely uniform distribution on $[-1,1]$ and Normal distribution.