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Entropy 2018, 20(9), 688; https://doi.org/10.3390/e20090688

Information Geometric Approach on Most Informative Boolean Function Conjecture

Department of Electronical and Electrical Engineering, Hongik University, Seoul 04066, Korea
Received: 26 July 2018 / Revised: 6 September 2018 / Accepted: 8 September 2018 / Published: 10 September 2018
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Abstract

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
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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No, A. Information Geometric Approach on Most Informative Boolean Function Conjecture. Entropy 2018, 20, 688.

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