{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:03:06Z","timestamp":1760241786125,"version":"build-2065373602"},"reference-count":12,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2018,9,10]],"date-time":"2018-09-10T00:00:00Z","timestamp":1536537600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002496","name":"Hongik University","doi-asserted-by":"publisher","award":["new faculty research support fund"],"award-info":[{"award-number":["new faculty research support fund"]}],"id":[{"id":"10.13039\/501100002496","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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  \u2192  { 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.<\/jats:p>","DOI":"10.3390\/e20090688","type":"journal-article","created":{"date-parts":[[2018,9,10]],"date-time":"2018-09-10T10:28:57Z","timestamp":1536575337000},"page":"688","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Information Geometric Approach on Most Informative Boolean Function Conjecture"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6346-4182","authenticated-orcid":false,"given":"Albert","family":"No","sequence":"first","affiliation":[{"name":"Department of Electronical and Electrical Engineering, Hongik University, Seoul 04066, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,9,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4515","DOI":"10.1109\/TIT.2014.2326877","article-title":"Which Boolean functions maximize mutual information on noisy inputs?","volume":"60","author":"Courtade","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Pichler, G., Matz, G., and Piantanida, P. (2016, January 11\u201314). A tight upper bound on the mutual information of two Boolean functions. Proceedings of the 2016 IEEE Information Theory Workshop (ITW), Cambridge, UK.","DOI":"10.1109\/ITW.2016.7606787"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ordentlich, O., Shayevitz, O., and Weinstein, O. (2016, January 10\u201315). An improved upper bound for the most informative Boolean function conjecture. Proceedings of the 2016 IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain.","DOI":"10.1109\/ISIT.2016.7541349"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4202","DOI":"10.1109\/TIT.2017.2686437","article-title":"On the optimal Boolean function for prediction under quadratic loss","volume":"63","author":"Weinberger","year":"2017","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Huleihel, W., and Ordentlich, O. (2017, January 25\u201330). How to quantize n outputs of a binary symmetric channel to n-1 bits?. Proceedings of the 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, Germany.","DOI":"10.1109\/ISIT.2017.8006496"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Nazer, B., Ordentlich, O., and Polyanskiy, Y. (2017, January 25\u201330). Information-distilling quantizers. Proceedings of the 2017 IEEE International Symposium on Information Theory (ISIT), Aachen, Germany.","DOI":"10.1109\/ISIT.2017.8006497"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/18.61115","article-title":"Divergence measures based on the Shannon entropy","volume":"37","author":"Lin","year":"1991","journal-title":"IEEE Trans. Inf. 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Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/9\/688\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:19:47Z","timestamp":1760195987000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/9\/688"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,9,10]]},"references-count":12,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2018,9]]}},"alternative-id":["e20090688"],"URL":"https:\/\/doi.org\/10.3390\/e20090688","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,9,10]]}}}