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A Note on the Entropy/Influence Conjecture University of Pennsylvania ScholarlyCommons Statistics Papers Wharton Faculty Research 11-28-2012 A Note on the Entropy/Influence Conjecture Nathan Keller Elchanan Mossel University of Pennsylvania Tomer Schlank Follow this and additional works at: https://repository.upenn.edu/statistics_papers Part of the Statistics and Probability Commons Recommended Citation Keller, N., Mossel, E., & Schlank, T. (2012). A Note on the Entropy/Influence Conjecture. Discrete Mathematics, 312 (22), 3364-3372. http://dx.doi.org/10.1016/j.disc.2012.07.031 At the time of publication, author Elchanan Mossel was affiliated with the University of California, Berkeley. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/statistics_papers/593 For more information, please contact [email protected]. A Note on the Entropy/Influence Conjecture Abstract The Entropy/Influence conjecture, raised by Friedgut and Kalai (1996) [9], seeks to relate two different measures of concentration of the Fourier coefficients of a Boolean function. Roughly ying,sa it claims that if the Fourier spectrum is “smeared out”, then the Fourier coefficients are concentrated on “high” levels. In this note we generalize the conjecture to biased product measures on the discrete cube, and prove a variant of the conjecture for functions with an extremely low Fourier weight on the “high” levels. Keywords entropy, influence, discrete fourier analysis, probabilistic combinatorics Disciplines Statistics and Probability Comments At the time of publication, author Elchanan Mossel was affiliated with the University of California, Berkeley. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania. This journal article is available at ScholarlyCommons: https://repository.upenn.edu/statistics_papers/593 A Note on the Entropy/Influence Conjecture Nathan Keller∗, Elchanan Mossel†, and Tomer Schlank‡ May 16, 2011 Abstract The entropy/influence conjecture, raised by Friedgut and Kalai [8] in 1996, seeks to relate two different measures of concentration of the Fourier coefficients of a Boolean function. Roughly saying, it claims that if the Fourier spectrum is “smeared out”, then the Fourier coefficients are concentrated on “high” levels. In this note we generalize the conjecture to biased product measures on the discrete cube, and prove a variant of the conjecture for functions with an extremely low Fourier weight on the “high” levels. 1 Introduction n Definition 1.1. Consider the discrete cube 0, 1 endowed with the product measure µp = n { } n n (pδ 1 + (1 p)δ 0 )⊗ , denoted in the sequel by 0, 1 p , and let f : 0, 1 p R. The Fourier- { } − { } { } { } → Walsh expansion of f with respect to the measure µp is the unique expansion f = αSuS, S 1,2,...,n ⊂{X } where for any T 1, 2, . , n ,1 ⊂ { } 1 p S T p S T u (T ) = − | ∩ | | \ |. S − p 1 p r r − In particular, for the uniform measure (i.e., p = 1/2), u (T ) = ( 1) S T . The coefficients α S − | ∩ | S are denoted by fˆ(S),2 and the level of the coefficient fˆ(S) is S . | | Properties of the Fourier-Walsh expansion are one of the main objects of study in discrete harmonic analysis. The entropy/influence conjecture, raised by Friedgut and Kalai [8] in 1996, seeks to relate two measures of concentration of the Fourier coefficients (i.e. coefficients of the Fourier-Walsh expansion) of Boolean functions. The first of them is the spectral entropy. ∗Weizmann Institute of Science. Partially supported by the Koshland Center for Basic Resarch. E-mail: [email protected]. †U.C. Berkeley and Weizmann Institute of Science. Supported by DMS 0548249 (CAREER) award, by DOD ONR grant N000141110140, by ISF grant 1300/08 and by a Minerva Grant. E-mail: [email protected]. ‡Hebrew University. Partially supported by the Hoffman program for leadership. E-mail: [email protected]. 1Throughout the paper, we identify elements of {0, 1}n with subsets of {1, 2, . , n} in the natural way. 2 Note that since the functions {uS }S⊂{1,...,n} form an orthonormal basis, the representation is indeed unique, ˆ and the coefficients are given by the formula f(S) = Eµp [f · uS ]. 1 Definition 1.2. Let f : 0, 1 n 1, 1 be a Boolean function. The spectral entropy of f { }p → {− } with respect to the measure µp is 2 1 Entp(f)= fˆ(S) log , ˆ 2 S 1,...,n f(S) ! ⊂{X } where the Fourier-Walsh coefficients are computed w.r.t. to µp. ˆ 2 Note that by Parseval’s identity, for any Boolean function we have S f(S) = 1, and thus, the squares of the Fourier coefficients can be viewed as a probability distribution on the set P 0, 1 n. In this notation, the spectral entropy is simply the entropy of this distribution, and { } intuitively, it measures how much are the Fourier coefficients “smeared out”. The second notion is the total influence. Definition 1.3. Let f : 0, 1 n 0, 1 . For 1 i n, the influence of the i-th coordinate { }p → { } ≤ ≤ on f with respect to µp is p Ii (f)= Pr [f(x) = f(x ei)], x µp ∼ 6 ⊕ where x e denotes the point obtained from x by replacing x with 1 x and leaving the other ⊕ i i − i coordinates unchanged. The total influence of the function f is n p Ip(f)= Ii (f). Xi=1 Influences of variables on Boolean functions were studied extensively in the last decades, and have applications in a wide variety of fields, including Theoretical Computer Science, Combinatorics, Mathematical Physics, Social Choice Theory, etc. (see, e.g., the survey [10].) As observed in [9], the total influence can be expressed in terms of the Fourier coefficients: Observation 1.4. Let f : 0, 1 n 1, 1 . Then { }p → {− } 1 I (f)= S fˆ(S)2. (1) p 4p(1 p) | | − XS In particular, for the uniform measure µ , I (f)= S fˆ(S)2. 1/2 1/2 S | | Thus, in terms of the distribution induced by theP Fourier coefficients, the total influence is (up to normalization) the expectation of the level of the coefficients, and it measures the question whether the coefficients are concentrated on “high” levels. The entropy/influence conjecture asserts the following: Conjecture 1.5 (Friedgut and Kalai). Consider the discrete cube 0, 1 n endowed with the { } uniform measure µ1/2. There exists a universal constant c, such that for any n and for any Boolean function f : 0, 1 n 1, 1 , { }1/2 → {− } Ent (f) c I (f). 1/2 ≤ · 1/2 2 The conjecture, if confirmed, has numerous significant implications. For example, it would imply that for any property of graphs on n vertices, the sum of influences is at least c(log n)2 (which is tight for the property of containing a clique of size log n). The best currently known 2 ǫ ≈ lower bound, by Bourgain and Kalai [5], is Ω((log n) − ), for any ǫ> 0. Another consequence of the conjecture would be an affirmative answer to a variant of a conjecture of Mansour [13] stating that if a Boolean function can be represented by a DNF formula of polynomial size in n (the number of coordinates), then most of its Fourier weight is concentrated on a polynomial number of coefficients (see [14] for a detailed explanation of this application). This conjecture, raised in 1995, is still wide open. In this note we explore the entropy/influence conjecture in two directions: Biased measure on the discrete cube. We state a generalization of the conjecture to the product measure µp on the discrete cube: Conjecture 1.6. There exists a universal constant c, such that for any 0 <p< 1, for any n and for any Boolean function f : 0, 1 n 1, 1 , { }p → {− } Ent (f) cp log(1/p) I (f). p ≤ · p We prove that Conjecture 1.6 follows from the original Entropy/Influence conjecture, and that it is tight for the graph property of containing a clique of fixed size (at the critical proba- bility). This answers a question raised by Kalai [11]. Functions with a low Fourier weight on the “high” levels. We consider a weaker version of the conjecture stating that if “almost all” the Fourier weight of a function is concentrated on the lowest k levels, then its entropy is at most c k. We prove this statement in an extreme · case: Proposition 1.7. Let f : 0, 1 n 1, 1 be a Boolean function such that all the Fourier { }1/2 → {− } weight of f is concentrated on the first k levels. Then all the Fourier coefficients of f are of the form k fˆ(S)= a(S) 2− · where a(S) Z. In particular, Ent (f) 2k. ∈ 1/2 ≤ Then we cite a stronger unpublished result of Bourgain and Kalai [6] which shows that if the Fourier weight beyond the kth level decays exponentially, then the spectral entropy is bounded from above by c k. · Finally, we suggest that if one could generalize the result of Bourgain and Kalai to some slower rate of decay, this would lead to a proof of the entire entropy/influence conjecture, using a tensorisation technique. This note is organized as follows. In Section 2 we consider the generalization of the en- tropy/influence conjecture to the biased measure on the discrete cube. Functions with a low Fourier weight on the high levels are discussed in Section 3. We conclude the paper with an easy proof of a weaker upper bound on the entropy, and with a connection between the entropy/influence conjecture and Friedgut’s characterization of functions with a low total in- fluence [7] in Section 4.
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