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Open AccessArticle

Information Geometric Approach on Most Informative Boolean Function Conjecture

Department of Electronical and Electrical Engineering, Hongik University, Seoul 04066, Korea
Entropy 2018, 20(9), 688; https://doi.org/10.3390/e20090688
Received: 26 July 2018 / Revised: 6 September 2018 / Accepted: 8 September 2018 / Published: 10 September 2018
Let X n be a memoryless uniform Bernoulli source and Y n be the output of it through a binary symmetric channel. Courtade and Kumar conjectured that the Boolean function f : { 0 , 1 } n { 0 , 1 } that maximizes the mutual information I ( f ( X n ) ; Y n ) is a dictator function, i.e., f ( x n ) = x i 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. View Full-Text
Keywords: Boolean function; Bregman divergence; clustering; geometric mean; Jensen–Shannon divergence Boolean function; Bregman divergence; clustering; geometric mean; Jensen–Shannon divergence
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No, A. Information Geometric Approach on Most Informative Boolean Function Conjecture. Entropy 2018, 20, 688.

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