{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article13365","name":"Improved Bounds on Fourier Entropy and Min-Entropy","abstract":"Given a Boolean function f:{-1,1}\u207f\u2192 {-1,1}, define the Fourier distribution to be the distribution on subsets of [n], where each S \u2286 [n] is sampled with probability f\u0302(S)\u00b2. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [E. Friedgut and G. Kalai, 1996] seeks to relate two fundamental measures associated with the Fourier distribution: does there exist a universal constant C>0 such that \u210d(f\u0302\u00b2)\u2264 C\u22c5 Inf(f), where \u210d(f\u0302\u00b2) is the Shannon entropy of the Fourier distribution of f and Inf(f) is the total influence of f?\r\nIn this paper we present three new contributions towards the FEI conjecture: \r\nii) Our first contribution shows that \u210d(f\u0302\u00b2) \u2264 2\u22c5 aUC^\u2295(f), where aUC^\u2295(f) is the average unambiguous parity-certificate complexity of f. This improves upon several bounds shown by Chakraborty et al. [S. Chakraborty et al., 2016]. We further improve this bound for unambiguous DNFs.\r\niii) We next consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture posed by O'Donnell and others [R. O'Donnell et al., 2011; R. O'Donnell, 2014] which asks if \u210d_{\u221e}(f\u0302\u00b2) \u2264 C\u22c5 Inf(f), where \u210d_{\u221e}(f\u0302\u00b2) is the min-entropy of the Fourier distribution. We show \u210d_{\u221e}(f\u0302\u00b2) \u2264 2\u22c5?_{min}^\u2295(f), where ?_{min}^\u2295(f) is the minimum parity certificate complexity of f. We also show that for all \u03b5 \u2265 0, we have \u210d_{\u221e}(f\u0302\u00b2) \u2264 2log (\u2016f\u0302\u2016_{1,\u03b5}\/(1-\u03b5)), where \u2016f\u0302\u2016_{1,\u03b5} is the approximate spectral norm of f. As a corollary, we verify the FMEI conjecture for the class of read-k DNFs (for constant k). \r\niv) Our third contribution is to better understand implications of the FEI conjecture for the structure of polynomials that 1\/3-approximate a Boolean function on the Boolean cube. We pose a conjecture: no flat polynomial (whose non-zero Fourier coefficients have the same magnitude) of degree d and sparsity 2^\u03c9(d) can 1\/3-approximate a Boolean function. This conjecture is known to be true assuming FEI and we prove the conjecture unconditionally (i.e., without assuming the FEI conjecture) for a class of polynomials. We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.","keywords":["Fourier analysis of Boolean functions","FEI conjecture","query complexity","polynomial approximation","approximate degree","certificate complexity"],"author":[{"@type":"Person","name":"Arunachalam, Srinivasan","givenName":"Srinivasan","familyName":"Arunachalam","email":"mailto:arunacha@mit.edu","affiliation":"MIT, Cambridge, MA, USA","funding":"Work done when at QuSoft, CWI, Amsterdam, supported by ERC Consolidator Grant 615307 QPROGRESS and MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert space."},{"@type":"Person","name":"Chakraborty, Sourav","givenName":"Sourav","familyName":"Chakraborty","email":"mailto:sourav@isical.ac.in","affiliation":"Indian Statistical Institute, Kolkata, India"},{"@type":"Person","name":"Kouck\u00fd, Michal","givenName":"Michal","familyName":"Kouck\u00fd","email":"mailto:koucky@iuuk.mff.cuni.cz","affiliation":"Computer Science Institute of Charles University, Prague, Czech Republic","funding":"Partially supported by ERC Consolidator Grant 616787 LBCAD, and GA\u010cR grant 19-27871X."},{"@type":"Person","name":"Saurabh, Nitin","givenName":"Nitin","familyName":"Saurabh","email":"mailto:nsaurabh@mpi-inf.mpg.de","affiliation":"Max Planck Institut f\u00fcr Informatik, Saarland Informatics Campus, Saarbr\u00fccken, Germany","funding":"Part of the work was done when the author was at IUUK, Prague and supported by the European Union\u2019s Seventh Framework Programme (FP\/2007-2013)\/ERC Grant Agreement no. 616787."},{"@type":"Person","name":"de Wolf, Ronald","givenName":"Ronald","familyName":"de Wolf","email":"mailto:rdewolf@cwi.nl","affiliation":"QuSoft, CWI and University of Amsterdam, the Netherlands","funding":"Partially supported by ERC Consolidator Grant 615307 QPROGRESS (ended Feb 2019) and by NWO under QuantERA project QuantAlgo 680-91-034 and the Quantum Software Consortium."}],"position":45,"pageStart":"45:1","pageEnd":"45:19","dateCreated":"2020-03-04","datePublished":"2020-03-04","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Arunachalam, Srinivasan","givenName":"Srinivasan","familyName":"Arunachalam","email":"mailto:arunacha@mit.edu","affiliation":"MIT, Cambridge, MA, USA","funding":"Work done when at QuSoft, CWI, Amsterdam, supported by ERC Consolidator Grant 615307 QPROGRESS and MIT-IBM Watson AI Lab under the project Machine Learning in Hilbert space."},{"@type":"Person","name":"Chakraborty, Sourav","givenName":"Sourav","familyName":"Chakraborty","email":"mailto:sourav@isical.ac.in","affiliation":"Indian Statistical Institute, Kolkata, India"},{"@type":"Person","name":"Kouck\u00fd, Michal","givenName":"Michal","familyName":"Kouck\u00fd","email":"mailto:koucky@iuuk.mff.cuni.cz","affiliation":"Computer Science Institute of Charles University, Prague, Czech Republic","funding":"Partially supported by ERC Consolidator Grant 616787 LBCAD, and GA\u010cR grant 19-27871X."},{"@type":"Person","name":"Saurabh, Nitin","givenName":"Nitin","familyName":"Saurabh","email":"mailto:nsaurabh@mpi-inf.mpg.de","affiliation":"Max Planck Institut f\u00fcr Informatik, Saarland Informatics Campus, Saarbr\u00fccken, Germany","funding":"Part of the work was done when the author was at IUUK, Prague and supported by the European Union\u2019s Seventh Framework Programme (FP\/2007-2013)\/ERC Grant Agreement no. 616787."},{"@type":"Person","name":"de Wolf, Ronald","givenName":"Ronald","familyName":"de Wolf","email":"mailto:rdewolf@cwi.nl","affiliation":"QuSoft, CWI and University of Amsterdam, the Netherlands","funding":"Partially supported by ERC Consolidator Grant 615307 QPROGRESS (ended Feb 2019) and by NWO under QuantERA project QuantAlgo 680-91-034 and the Quantum Software Consortium."}],"copyrightYear":"2020","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2020.45","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["http:\/\/arxiv.org\/abs\/1411.5729","http:\/\/arxiv.org\/abs\/1306.3362","http:\/\/arxiv.org\/abs\/1809.09819","http:\/\/arxiv.org\/abs\/quant-ph\/9802049","http:\/\/arxiv.org\/abs\/1605.07084","http:\/\/arxiv.org\/abs\/0911.4007","http:\/\/arxiv.org\/abs\/1909.10428","http:\/\/arxiv.org\/abs\/1110.4301","http:\/\/arxiv.org\/abs\/0904.3540","http:\/\/arxiv.org\/abs\/1604.07432","http:\/\/arxiv.org\/abs\/1711.00762","https:\/\/terrytao.wordpress.com\/2007\/08\/16\/gil-kalai-the-entropyinfluence-conjecture\/","http:\/\/arxiv.org\/abs\/1105.2651","http:\/\/arxiv.org\/abs\/1208.0161","http:\/\/arxiv.org\/abs\/0909.3392","http:\/\/arxiv.org\/abs\/1304.1347","http:\/\/arxiv.org\/abs\/1312.2143","http:\/\/arxiv.org\/abs\/1512.01603","http:\/\/arxiv.org\/abs\/1604.07595","http:\/\/eccc.hpi-web.de\/report\/2012\/144","http:\/\/arxiv.org\/abs\/1806.03646","http:\/\/arxiv.org\/abs\/1304.1245","http:\/\/arxiv.org\/abs\/1312.3003","http:\/\/arxiv.org\/abs\/1307.6738"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6357","volumeNumber":154,"name":"37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)","dateCreated":"2020-03-04","datePublished":"2020-03-04","editor":[{"@type":"Person","name":"Paul, Christophe","givenName":"Christophe","familyName":"Paul","email":"mailto:christophe.paul@lirmm.fr","sameAs":"https:\/\/orcid.org\/0000-0001-6519-975X","affiliation":"CNRS, Universit\u00e9 de Montpellier, France"},{"@type":"Person","name":"Bl\u00e4ser, Markus","givenName":"Markus","familyName":"Bl\u00e4ser","email":"mailto:mblaeser@cs.uni-saarland.de","affiliation":"Universit\u00e4t des Saarlandes, Saarbr\u00fccken, Germany"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article13365","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6357"}}}