A Differential Equation Approach to the Most-Informative Boolean Function Conjecture

TL;DR

This paper introduces a differential equation framework to analyze the most-informative Boolean function conjecture, formulates related inequalities, and proposes a specific inequality whose proof could resolve the conjecture.

Contribution

It develops a novel differential equation approach and formulates a key finite-dimensional inequality that could prove the conjecture in the balanced case.

Findings

Formulated a functional inequality on finite-dimensional random variables.

Developed a similar inequality for the Hellinger conjecture.

Proposed a specific finite-dimensional inequality that, if proved, would resolve the conjecture.

Abstract

We study the most-informative Boolean function conjecture using a differential equation approach. This leads to a formulation of a functional inequality on finite-dimensional random variables. We also develop a similar inequality in the case of the Hellinger conjecture. Finally, we conjecture a specific finite dimensional inequality that, if proved, will lead to a proof of the Boolean function conjecture in the balanced case. We further show that the above inequality holds modulo four explicit inequalities (all of which seems to hold via numerical simulation) with the first three containing just two variables and a final one involving four variables.