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An Improved Upper Bound for the Most Informative Boolean Function Conjecture [article]

Or Ordentlich, Ofer Shayevitz, Omri Weinstein
2015 arXiv   pre-print
In this paper, we derive a new upper bound that holds for all balanced functions, and improves upon the best known bound for all 13<α<12.  ...  A recent conjecture by Courtade and Kumar postulates that I(f(X);Y)≤ 1-h(α) for any Boolean function f. So far, the best known upper bound was I(f(X);Y)≤ (1-2α)^2.  ...  upper bound on the mutual information I(f (X); Y ), that holds for any balanced Boolean function.  ... 
arXiv:1505.05794v2 fatcat:ujk2bix7zzd5njkauifx72vwie

Improved bounds on Fourier entropy and Min-entropy [article]

Srinivasan Arunachalam and Sourav Chakraborty and Michal Koucký and Nitin Saurabh and Ronald de Wolf
2021 arXiv   pre-print
This improves upon Chakraborty et al. An important consequence of the FEI conjecture is the long-standing Mansour's conjecture.  ...  Given a Boolean function f:{-1,1}^n→{-1,1}, the Fourier distribution assigns probability f(S)^2 to S⊆ [n].  ...  New upper bounds for the FMEI conjecture Given the hardness of obtaining better upper bounds on the Fourier entropy of a Boolean function, we make progress on a weaker conjecture, the FMEI conjecture.  ... 
arXiv:1809.09819v2 fatcat:xdvzfqhxzbfubndtq3hudfrkey

Improved Bounds on Fourier Entropy and Min-entropy

Srinivasan Arunachalam, Sourav Chakraborty, Michal Koucký, Nitin Saurabh, Ronald De Wolf
2021 ACM Transactions on Computation Theory  
This improves upon several bounds shown by Chakraborty et al. [20]. We further improve this bound for unambiguous DNFs.  ...  We discuss an intriguing connection between our conjecture and the constant for the Bohnenblust-Hille inequality, which has been extensively studied in functional analysis.  ...  Conclusion We gave improved upper bounds on Fourier entropy of Boolean functions in terms of average unambiguous (parity)-certificate complexity, and as a corollary verified the FEI conjecture for functions  ... 
doi:10.1145/3470860 fatcat:fsbv5cpcindptc6puqczud22me

Degree and Sensitivity: tails of two distributions [article]

Parikshit Gopalan, Rocco Servedio, Avishay Tal, Avi Wigderson
2016 arXiv   pre-print
Further, we show that improving the bound to O(s^c log(1/epsilon)^d) for any d < 1 and any c > 0 will imply the sensitivity conjecture.  ...  We postulate a robust analogue of the sensitivity conjecture: if most inputs to a Boolean function f have low sensitivity, then most of the Fourier mass of f is concentrated on small subsets, and present  ...  We also thank Yuval for useful discussions, and thank David Levin and Yuval Peres for letting us present the proof of Lemma 4.12 here.  ... 
arXiv:1604.07432v1 fatcat:rb7puf3tbfftphxyou4tc2karu

Communication is bounded by root of rank [article]

Shachar Lovett
2013 arXiv   pre-print
Equivalently, any graph whose adjacency matrix has rank r has chromatic number at most 2^O(√(r)·(r)). This gives a nearly quadratic improvement in the dependence on the rank over previous results.  ...  We prove that any total boolean function of rank r can be computed by a deterministic communication protocol of complexity O(√(r)·(r)).  ...  Acknowledgements I thank Dmitry Gavinsky, Pooya Hatami, Russell Impagliazzo and Adi Shraibman for helpful discussions, and Salil Vadhan for allowing to present his simplified proof of Lemma 3.1.  ... 
arXiv:1306.1877v2 fatcat:gjwhtfbefvc5bko63plkn2yqui

Extremal properties of polynomial threshold functions

Ryan O'Donnell, Rocco A. Servedio
2008 Journal of computer and system sciences (Print)  
In this paper we give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following:  ...  Acknowledgment We would like to thank Johan Håstad for telling us the proof of Theorem 11 and letting us reproduce it here.  ...  One goal is to improve the lower order term in our n/2 + O( √ n log n ) upper bound for the PTF degree of almost every Boolean function.  ... 
doi:10.1016/j.jcss.2007.06.021 fatcat:437jjuo2qjapteir6vynsxsn7y

Improved Composition Theorems for Functions and Relations

Sajin Koroth, Or Meir, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization  
As an approach for this question, Karchmer, Raz and Wigderson [5] proposed a conjecture called the KRW conjecture, which if true, would imply that P is not cotained in N C 1 .  ...  In this work we significantly improve the parameters in these variants, achieving almost tight lower bounds.  ...  The KRW conjecture says that the upper bound is essentially optimal for KW f g .  ... 
doi:10.4230/lipics.approx-random.2018.48 dblp:conf/approx/KorothM18 fatcat:sapv2l5y45hedfbcaho2qr7cgm

A Little Advice Can Be Very Helpful [chapter]

Arkadev Chattopadhyay, Jeff Edmonds, Faith Ellen, Toniann Pitassi
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
We also prove lower bounds in some restricted versions of this model for natural functions such as setdisjointness and inner product. All of our upper bounds conform to these restrictions.  ...  We show that there exist Boolean functions requiring linear randomized communication complexity in the two-party model, for which the asymmetric versions in his model have deterministic protocols with  ...  The first author is also supported by a postdoctoral fellowship of the Ontario Ministry of Research and Innovation.  ... 
doi:10.1137/1.9781611973099.52 dblp:conf/soda/ChattopadhyayEEP12 fatcat:sz2qxfoiarcfrgnalz4ultiyd4

Ordered dynamics in biased and cooperative Boolean networks

Winfried Just, Germán Enciso
2013 Advances in Difference Equations  
For the case of n-dimensional networks with r = 2 in which only AND and OR are allowed, we find an upper bound of 10 n/4 , which is asymptotically optimal in view of previously published counterexamples  ...  We prove nontrivial upper bounds on the maximum length of periodic orbits in such networks under the assumption that the maximum number of inputs and outputs per node is a fixed constant r.  ...  Acknowledgements We would like to thank the referees for insightful and valuable suggestions on how to improve the manuscript.  ... 
doi:10.1186/1687-1847-2013-313 fatcat:jjxya6zbsrh3pnt3hh6dzqteqm

On the Sensitivity Conjecture for Disjunctive Normal Forms

Karthik C. S., Sébastien Tavenas, Marc Herbstritt
2016 Foundations of Software Technology and Theoretical Computer Science  
The sensitivity conjecture of Nisan and Szegedy [CC'94] asks whether for any Boolean function f , the maximum sensitivity s(f ), is polynomially related to its block sensitivity bs(f ), and hence to other  ...  We extend this result and also construct examples of Boolean functions which provide the matching lower bounds.  ...  Finally, we would like to thank the anonymous reviewers for helping us improve the presentation of the paper.  ... 
doi:10.4230/lipics.fsttcs.2016.15 dblp:conf/fsttcs/ST16 fatcat:6nxtzgtssvegnej7pcxqpc7mv4

Variations on the Sensitivity Conjecture [article]

Pooya Hatami, Raghav Kulkarni, Denis Pankratov
2010 arXiv   pre-print
We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open.  ...  Just as Laci seemed to give up on popularizing the conjecture, Sasha Razborov rekindled the flame. We would like to thank Sasha for introducing us to the Sensitivity Conjecture.  ...  to the Sensitivity Conjecture, including David Rubinstein, Sandy Kutin, and Sourav Chakraborty.  ... 
arXiv:1011.0354v1 fatcat:wlwbaxnk5rgw5ab5eoxiwlphra

Page 7002 of Mathematical Reviews Vol. , Issue 2002I [page]

2002 Mathematical Reviews  
For the perfect non- linear S-boxes designed for block ciphers, an upper bound on the maximum correlation coefficients is presented.”  ...  However, S-boxes tend to leak more information about the LFSR sequences than Boolean functions.  ... 

Candidate Boolean Functions towards Super-Quadratic Formula Size

Kenya UENO
2015 IEICE transactions on information and systems  
First, we consider recursive Boolean functions and prove their general formula size upper bounds. We also discuss recursive Boolean functions based on exact 2-bit functions.  ...  In particular, we discuss the structure of an optimal protocol partition for the Karchmer-Wigderson communication game. key words: Boolean function, computational complexity, formula complexity  ...  Acknowledgments The author is grateful to anonymous referees for their helpful comments to improve the presentation of the paper.  ... 
doi:10.1587/transinf.2014fcp0011 fatcat:vdirtm3jkrhonofaoohtcvnho4

Spectral properties of threshold functions

Craig Gotsman, Nathan Linial
1994 Combinatorica  
Some conjectures are posed concerning upper and lower bounds on influences of variables in higher order threshold functions.  ...  We examine the spectra of boolean functions obtained as the sign of a real polynomial of degree d.  ...  We would like to thank Mario Szegedy and Ilan Newman for interesting discussions during this work.  ... 
doi:10.1007/bf01305949 fatcat:vvoryjqtebewhctokzurdrq5w4

On the Most Informative Boolean Functions of the Very Noisy Channel [article]

Hengjie Yang, Richard D. Wesel
2019 arXiv   pre-print
Along the way, we show that the dictator function is the most informative function in the high noise regime.  ...  A recent conjecture postulated by Courtade and Kumar states that for any Boolean function f:{0,1}^n→{0,1}, I(f(X^n);Y^n)< 1-H(α).  ...  In [4] , Ordentlich, Shayevitz, and Weinstein used Fourier analytic techniques and leveraged hypercontractivity to improve the upper bound on I(f (X n ); Y n ) for all balanced Boolean functions, i.e.  ... 
arXiv:1807.11289v3 fatcat:6z4wp2pv7jdlxklyad2rm6nrze
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