On The Most Discriminative Boolean Functions for Correlated Sources

Motivated by a conjecture of Amari and Kobayashi, we study the problem of identifying pairs of Boolean functions that maximize the Kullback-Leibler divergence between two distributions obtained by separately compressing two correlated sources. When the reference distribution corresponds to independent sources, this problem reduces to the problem of maximizing mutual information, for which the optimality of dictator functions has been proved by Pichler, Piantanida, and Matz. For the problem of maximizing Fisher information, which can be viewed as a local version of the problem studied in this paper, Amari and Kobayashi conjectured that parity functions are optimal. For unbiased pairs of Boolean functions, and for identical pairs in the nonnegative correlation regime, we prove that both the divergence and the Fisher information are maximized by level-$k$ functions, namely, functions whose Fourier coefficients are supported only on level $k$. Since level-$k$ functions include parity functions, this gives a partial resolution of the conjecture of Amari and Kobayashi. Furthermore, in the framework of Bayesian distributed one-bit hypothesis testing, we prove that level-$k$ functions are optimal among all pairs of functions. Finally, we also discuss the one function version of the problem studied in this paper, which can be regarded as the divergence analogue of the Courtade and Kumar conjecture.