A Differential Equation Approach to the Most-Informative Boolean Function Conjecture
Zijie Chen, Amin Gohari, Chandra Nair
TL;DR
This work introduces a differential-equation framework that traces a continuum of degraded channels $\boldsymbol{Y}_t$ with crossover $p_t=(1-e^{-2t})/2$ to bound information-theoretic quantities for the most-informative Boolean function conjecture and its Hellinger variant. Central to the approach is a finite-dimensional functional inequality defined via a class of nonnegative functions $\psi\in\Psi$ (and its Hellinger counterpart $\Psi_H$), which yields derivative bounds on $\gamma(t)=H(F|\boldsymbol{Y}_t)$ and $r(t)=\mathbb{E}\sqrt{1-d_{oldsymbol{X}}(t)^2}$; solving these yields dimension-free lower bounds on the target information measures. A key conjecture links a four-variable infimum $\zeta$ to a function in $\Psi$ (via $\phi$), and, if true, would imply the balanced-case MI conjecture. The paper provides rigorous proofs for the minimization framework (existence, reduction to structured mixtures) and demonstrates extensive numerical support for the conjectures, offering a novel analytic route to resolving long-standing questions at the intersection of information theory and isoperimetric inequalities.
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.