A Low-Depth Monotone Function that is not an Approximate Junta

TL;DR

This paper presents a monotone Boolean function with a low-depth decision tree that cannot be closely approximated by small juntas, challenging assumptions about the simplicity of such functions.

Contribution

It introduces a specific example of a low-depth monotone function that defies approximation by small juntas, highlighting limitations in existing approximation techniques.

Findings

Low-depth monotone functions can be complex and not approximable by small juntas.

The example challenges previous beliefs about the simplicity of low-depth monotone functions.

Implications for the analysis of Boolean functions and complexity theory.

Abstract

We provide an example of a monotone Boolean function on the hypercube given by a low depth decision tree that is not well approximated by any k-junta for small k.