Information Geometric Approach on Most Informative Boolean Function Conjecture

Abstract

Let be a memoryless uniform Bernoulli source and be the output of it through a binary symmetric channel. Courtade and Kumar conjectured that the Boolean function that maximizes the mutual information is a dictator function, i.e., for some i. We propose a clustering problem, which is equivalent to the above problem where we emphasize an information geometry aspect of the equivalent problem. Moreover, we define a normalized geometric mean of measures and interesting properties of it. We also show that the conjecture is true when the arithmetic and geometric mean coincide in a specific set of measures.