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

2025-02-14

AI Summary

This work addresses the “Most Informative Boolean Function Conjecture,” a central open problem in discrete information theory. We introduce, for the first time, a differential-equation-based approach to reformulate this infinite-dimensional information-theoretic problem as a functional inequality over finite-dimensional random variables. Our contributions are threefold: (1) We derive a novel information inequality corresponding to the Hellinger conjecture; (2) we develop a verifiable finite-dimensional reduction framework, equivalently transforming the original conjecture into four explicit low-dimensional inequalities—three bivariate and one quadrivariate; (3) we rigorously prove the equivalence of the reduction and numerically verify all resulting inequalities. This yields the first computationally tractable and falsifiable finite-dimensional criterion for the conjecture in the balanced case. Moreover, our methodology pioneers a synergistic use of differential analysis and probabilistic inequalities to tackle discrete information-theoretic conjectures.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Probabilistic Inference

Application Category

• Web Mining and Content Analysis: Content-based information diffusion
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Study most-informative Boolean function conjecture
• Formulate functional inequality on random variables
• Conjecture specific finite-dimensional inequality

Innovation

• Differential equation approach
• Functional inequality formulation
• Numerical simulation verification