The Entropy Influence Conjecture Revisited

The En trop y Influence Conjecture Revisited Bireswar Das ∗ Manjish P a l † Vijay Visav aliya ‡ Abstract In this pap er, w e pro ve that most of the b o olean functions, f : {− 1 , 1 } n → {− 1 , 1 } satisfy the F ourier En trop y Infl uence (FEI) Conjecture due to F riedgut and Kalai (Pro c. AMS’96)[1]. The conjecture sa ys that the En tr op y of a b o olean function is at most a constan t times th e Infl uence o f the function. The conjecture has b een pro v en for families of functions of sm aller sizes. O’donnell, W right and Zhou (ICALP’11)[7] v erifi ed the conjecture for the family of s ymmetric fun ctions, w hose size is 2 n +1 . They are in f act able to prov e the conjecture for the family of d -part sym m etric fun ctions for constan t d , the size of whose is 2 O ( n d ) . Also it is known that the conjecture is true for a large fraction of p olynomial sized DNFs (COL T’10)[5]. Usin g elementa ry m etho d s w e pro v e that a ran d om function with high probabilit y satisfies th e conjecture with the constan t as (2 + δ ), for any constan t δ > 0. 1 In tro duct i o n The Entrop y Influence Conjecture due t o F riedgut and Kalai [1] says that for ev ery b o olean function f : {− 1 , 1 } n → {− 1 , 1 } the fo llo wing holds, X S ⊆ [ n ] b f ( S ) 2 log 2 1 b f ( S ) 2 ≤ C · X S ⊆ [ n ] b f ( S ) 2 | S | for some unive rsal constan t C > 0 where b f ( S ) is the co efficien t of χ S ( x ) in the F ourier expansion of f . The conj ecture is o f profo und imp ortance b ecause of its p oten tial impacts in areas lik e Learning Theory , Threshold Phenomena in monotone graph prop erties, metric em b eddings etc. F or a detailed des cription of the impact and bac kground of the FEI conjec- ture the reader is recommended to read the In tro duction section of [7] and the blog po st b y Gil Kalai [3 ]. The conjecture can b e ve rified for simple functions lik e AND, OR, MAJORITY, T rib es etc. Although p osed abo ut 15 y ears bac k w e are not a w are of a signific an tly large (of ∗ Indian Institute o f T ec hnology Gandhinagar , India. bireswar @iitgn.ac .in † Indian Institute o f T ec hnology Gandhinagar , India. manjish pal@ii tgn.ac .in ‡ Vishw ak ar ma Gov ernment Engineering College Gandhinaga r, India. visaval iavija y@gmail.com 1 doubly exp onen t ial size) fa mily of functions whic h satisfies this conjecture. Kliv ans et a l. [5] pro ved recen tly that a large fraction of p olynomial sized DNF formulae satisfy the Mansour’s conjecture [6] whic h in turn implies that FEI conjecture is also true for those functions, a class whic h has a size of 2 poly ( n ) . In a recen t resurrection o f the FEI conjecture, O’Donnell et al. [7] pro v ed the conjecture for the fa mily of sy mmetric functions and d -part symm etric functions for constan t d . The sizes of these fa milies a re 2 n +1 and 2 O ( n d ) resp ectiv ely , again only exp onen t ia l in size. They a lso v erified it for read-once decision trees whic h are of exp o nen tial size as w ell. Th us, one is not aw are of an explicit or non-explicit family of doubly expo nen tial size that satisfies the conjecture. V ery recen t ly , in a note, Keller et al [4] managed to prov e a v ariant of the conjecture fo r functions whic h ha v e low F ourier w eigh t o n c haracters of large size. 2 Preliminaries In t his section we introduce the basic preliminaries of discrete F ourier Analysis whic h will b e of in terest for us. Definition 1. L et f : {− 1 , 1 } n → {− 1 , 1 } b e a b o ole an function. The F ourier expansion of f is written as f ( x ) = X S ⊆ [ n ] b f ( S ) χ S ( x ) wher e χ S ( x ) = Π i ∈ S x i . Definition 2. L et f : {− 1 , 1 } n → {− 1 , 1 } b e a b o ole an function. The en trop y of f is define d as H ( f ) = X S ⊆ [ n ] b f ( S ) 2 log 2 1 b f ( S ) 2 Notice b y Parse v al’s identit y , P S ⊆ [ n ] b f ( S ) 2 = 1 whic h implies that the F ourier co efficien t s can be though t of as a probabilit y distribution and hence H ( f ) giv es us the en tro py o f that distribution. The follo wing fact (see page 40 of [2]) giv es an upp er b ound on H ( f ) whic h will be of use for us later. F act 1. F or an arbi tr ary b o ole an function f , H ( f ) ≤ n . Definition 3. F or a b o ole an function f : {− 1 , 1 } n → {− 1 , 1 } , Inf i ( f ) , the Influence of co ordinate i is d efine d as Inf i ( f ) = X S : i ∈ S b f ( S ) 2 and the Influence of f , Inf ( f ) , is define d as Inf ( f ) = n X i =1 Inf i = X S ⊆ [ n ] b f ( S ) 2 | S | 2 An equiv alen t com binatorial in terpretation of the influence of t he i th co ordinate is g iven b y Inf i = Pr x [ f ( x ) 6 = f ( x ( i ) )] whe re x ( i ) is x with the i th co ordinate flipp ed. W e will also use the follow ing w ell-know n facts regarding the F ourier co efficien ts. F act 2. F or a b o ol e an function f : {− 1 , 1 } n → {− 1 , 1 } , the fol lowing holds for any subset S ⊆ [ n ] 1. b f ( S ) = 1 2 n X x f ( x ) χ S ( x ) . 2. X x χ S ( x ) = 0 if S 6 = φ , and 2 n if S = φ . 3 The Result The main contribution of our pap er is to prov e that there is a large (non-explicit) family of functions that satisfies the en trop y influence conjecture. The size of this family is significantly larger t ha n t he size of an y of the kno wn families for whic h the FEI conjecture is kno wn to b e true. More precisely , w e show tha t there is a family of functions whose size is " 1 − 4  1 + 2 δ  2 1 2 n +1 n # · 2 2 n satisfies the conjecture with C = 2 + δ for a n y constan t δ > 0. Of course, o ur r esult do esn’t pro vide a step tow ards the resolving the FEI conjecture as it is kno wn that t he there are functions whic h need the constan t C to b e at least 4.6 [7]. Theorem 1. A r andom function satisfies the FEI c o n je ctur e with hig h pr ob ability with C = 2 + δ , for any c onstant δ > 0 . Consider ra ndom function whic h puts v alues 1 or − 1 indep enden tly on ev ery p oin t of { 1 , − 1 } n with an equal probability of 1 / 2. Clearly , ev ery function is obtained with a prob- abilit y of 1 2 2 n . Let H r and I r b e random v ariables denoting t he en tro p y and influence of a randomly c hosen function as ab ov e. W e will pro ve T heorem 1 using a simple a pplication of Cheb yshev Inequalit y . W e will use the follo wing t wo lem mas in our proo f . Lemma 1. E [ I r ] = n 2 . 3 Pr o of. E [ I r ] = E   X S ⊆ [ n ] b f ( S ) 2 | S |   = X S ⊆ [ n ] E [ b f ( S ) 2 ] | S | = X S ⊆ [ n ] 1 2 2 n E " X x f ( x ) 2 χ S ( x ) 2 + X x 6 = y f ( x ) f ( y ) χ S ( x ) χ S ( y ) # | S | = 1 2 2 n X S ⊆ [ n ] X x 1 + X x 6 = y E [ f ( x ) f ( y )] χ S ( x ) χ S ( y ) ! | S | = 1 2 2 n X S ⊆ [ n ] 2 n | S | + 0 (this follo ws because for x 6 = y , E [ f ( x ) f ( y )] = E [ f ( x )] E [ f ( y )] = 0 ) = 1 2 n n X k =0  n k  k = n 2 n − 1 2 n = n 2 Lemma 2. V ar [ I r ] = n 2 n +1 . Pr o of. W e hav e already calculated E [ I r ]. T o calculate, V ar [ I r ] = E [ I 2 r ] − ( E [ I r ]) 2 w e need to calculate E [ I 2 r ]. E [ I 2 r ] = E     X S ⊆ [ n ] b f ( S ) 2 | S |   2   = E   X S 1 ,S 2 ⊆ [ n ] b f ( S 1 ) 2 b f ( S 2 ) 2 | S 1 || S 2 |   = X S 1 ,S 2 ⊆ [ n ] E h b f ( S 1 ) 2 b f ( S 2 ) 2 i | S 1 || S 2 | 4 E h b f ( S 1 ) 2 b f ( S 2 ) 2 i = 1 2 4 n E   X x f ( x ) χ S 1 ( x ) ! 2 X x f ( x ) χ S 2 ( x ) ! 2   = 1 2 4 n E " X x 1 ,y 1 ,x 2 ,y 2 f ( x 1 ) f ( y 1 ) f ( x 2 ) f ( y 2 ) χ S 1 ( x 1 ) χ S 1 ( y 1 ) χ S 2 ( x 2 ) χ S 2 ( y 2 ) # = 1 2 4 n X x 1 ,y 1 ,x 2 ,y 2 E [ f ( x 1 ) f ( y 1 ) f ( x 2 ) f ( y 2 )] χ S 1 ( x 1 ) χ S 1 ( y 1 ) χ S 2 ( x 2 ) χ S 2 ( y 2 ) T o calculate the ab o ve sum, consider the follo wing sets, A 1 = { ( x 1 , y 1 , x 2 , y 2 ) | x 1 = y 1 , x 2 = y 2 } A 2 = { ( x 1 , y 1 , x 2 , y 2 ) | x 1 = x 2 , y 1 = y 2 } A 3 = { ( x 1 , y 1 , x 2 , y 2 ) | x 1 = y 2 , x 2 = y 1 } Notice the follo wing prop erties of A 1 , A 2 , A 3 , | A 1 | = | A 2 | = | A 3 | = 2 2 n , A 1 ∩ A 2 = A 2 ∩ A 3 = A 3 ∩ A 1 = A 1 ∩ A 2 ∩ A 3 and | A 1 ∩ A 2 ∩ A 3 | = 2 n . It is easy to v erify that if ( x 1 , y 1 , x 2 , y 2 ) / ∈ A 1 S A 2 S A 3 , then E [ f ( x 1 ) f ( y 1 ) f ( x 2 ) f ( y 2 )] = 0. Otherwise it is χ S 1 ( x 1 ) χ S 1 ( y 1 ) χ S 2 ( x 2 ) χ S 2 ( y 2 ). Using the ab ov e prop erties a nd inclusion exclusion principle w e ha v e E h b f ( S 1 ) 2 b f ( S 2 ) 2 i equal to 1 2 4 n " X A 2 χ S 1 ( x 1 ) χ S 1 ( y 1 ) χ S 2 ( x 2 ) χ S 2 ( y 2 ) + X A 3 χ S 1 ( x 1 ) χ S 1 ( y 1 ) χ S 2 ( x 2 ) χ S 2 ( y 2 ) # + 1 2 4 n " X A 1 χ S 1 ( x 1 ) χ S 1 ( y 1 ) χ S 2 ( x 2 ) χ S 2 ( y 2 ) # − 2 · 1 2 4 n " X A 1 ∩ A 2 ∩ A 3 χ S 1 ( x 1 ) χ S 1 ( y 1 ) χ S 2 ( x 2 ) χ S 2 ( y 2 ) # = 1 2 4 n " X x,y χ S 1 ( x ) χ S 1 ( y ) χ S 2 ( x ) χ S 2 ( y ) + X x,y χ S 1 ( x ) χ S 1 ( y ) χ S 2 ( y ) χ S 2 ( x ) # + 1 2 4 n " X x,y χ S 1 ( x ) χ S 1 ( x ) χ S 2 ( y ) χ S 2 ( y ) # − 2 · 1 2 4 n · 2 n = 1 2 4 n " 2 X x,y χ S 1 ∆ S 2 ( x ) χ S 1 ∆ S 2 ( y ) + 2 2 n − 2 · 2 n # . Th us, using the fact that P x χ S 1 ∆ S 2 ( x ) = 0 if S 1 6 = S 2 and 2 n otherwise, w e ha v e E h b f ( S 1 ) 2 b f ( S 2 ) 2 i =  1 2 4 n · (2 2 n − 2 · 2 n ) if S 1 6 = S 2 . 1 2 4 n · (3 · 2 2 n − 2 · 2 n ) otherwise. 5 Therefore, E [ I 2 r ] = X S 1 ,S 2 ⊆ [ n ] E h b f ( S 1 ) 2 b f ( S 2 ) 2 i | S 1 || S 2 | = X S 1 = S 2 E h b f ( S 1 ) 2 b f ( S 2 ) 2 i | S 1 || S 2 | + X S 1 6 = S 2 E h b f ( S 1 ) 2 b f ( S 2 ) 2 i | S 1 || S 2 | = X S 1 = S 2 3 · 2 2 n − 2 · 2 n 2 4 n | S 1 || S 2 | + X S 1 6 = S 2 2 2 n − 2 · 2 n 2 4 n | S 1 || S 2 | = X S 1 = S 2 2 · 2 2 n 2 4 n | S 1 || S 2 | + X S 1 ,S 2 2 2 n − 2 · 2 n 2 4 n | S 1 || S 2 | = 2 · 2 2 n 2 4 n n X k =0  n k  k 2 + 2 2 n − 2 · 2 n 2 4 n n X k =0  n k  k ! 2 = 2( n ( n − 1)2 n − 2 + n 2 n − 1 ) 2 2 n +  1 2 2 n − 2 2 3 n  ( n 2 n − 1 ) 2 = n 2 n +1 + n 2 4 . Hence, V ar [ I r ] = n 2 n +1 . W e are no w ready to pro v e Theorem 1. Pr o of. ( Theorem 1) W e will pro v e this using simple applications of Cheby shev inequalit y . As mentioned earlier w e pic k a ra ndom function whic h puts v alues − 1 or 1 independen t ly on eve ry p oin t o f {− 1 , 1 } n with an equal probability of 1 / 2 . Recall that, H r and I r are random v aria bles denoting the en trop y and influence of a randomly c hosen function. Let us define the ev ent E F E I indicating that the ra ndomly chosen b o olean function satisfies the FEI conjecture with the constan t C = 2 + 2 ǫ for ǫ > 0. More precisely , E F E I is the ev en t that H r ≤ (2 + 2 ǫ ) I r . Our a im is to pro ve that E F E I o ccurs with high proba bility . F ro m F act 1, w e hav e H r ≤ n . Therefore, Pr[ H r > n ] = 0 . Consider the follo wing ev en ts: E 1 := H r > n E 2 := (2 + 2 ǫ ) I r ≤ n No w if H r > (2 + 2 ǫ ) I r , then either E 1 or E 2 happ ens. Therefore b y union b ound, Pr[ H r > (2 + 2 ǫ ) I r ] ≤ Pr[ E 1 ] + Pr[ E 2 ] 6 Since from Lemma 1, E [ I r ] = n/ 2, E [(2 + 2 ǫ ) I r ] = (1 + ǫ ) n . No w we upp er b ound Pr[ E 2 ]. Pr[ E 2 ] = Pr[(2 + 2 ǫ ) I r ≤ n ] = Pr[(2 + 2 ǫ ) I r − (1 + ǫ ) n ≤ − ǫn ] ≤ Pr[ | (2 + 2 ǫ ) I r − (1 + ǫ ) n | ≥ ǫn ] Using Cheb yshev Inequalit y , Pr[ | (2 + 2 ǫ ) I r − (1 + ǫ ) n | ≥ ǫn ] ≤ V ar [(2 + 2 ǫ ) I r ] ǫ 2 n 2 = 4  1 + 1 ǫ  2 V ar [ I r ] n 2 . F rom Lemma 2, t his in turn implies Pr[ H r > (2 + 2 ǫ ) I r ] ≤ 4  1 + 1 ǫ  2 1 2 n +1 n Hence, it follows that Pr[ E F E I ] ≥ 1 − 2  1 + 1 ǫ  2 1 2 n n with constan t C = 2 + 2 ǫ for arbitra r y constan t ǫ > 0 . 4 Conclus ion In this pap er w e ga ve a simple pro of o f the fact a random function will almost surely satisfy the FEI conjecture for C = 2 + δ for δ > 0. Although our pro of is non-constructiv e, this is the only doubly exponential sized family for which it is know n that the FEI conjecture is true. It w ould be in teresting to get a large ( ω (2 poly ( n ) )) explicit family o f functions that satisfy FEI conjecture. One p ossible candidate is the class of functions f ( x 1 , x 2 , . . . , x p ) whic h are in v ariant under the action of the cyclic p erm uta tion gro up C p ≤ Sym( p ) where p is prime. It can be v erified that the size of this class is 2 2 p − 2 p +2 . Because of the highly structured nature of the functions whic h are in v ariant under C p it migh t b e plausible to v erify the conjecture for these functions. 5 Ac kno wledg emen t W e thank K unal Dutta and Justin Salez for p oin ting o ut that our res ult can b e ex tended to a high probabilit y statemen t. References [1] E. F riedgut and G. Kalai. Every mon o tone gr aph pr op erty has a sharp thr eshold , Pro- ceedings of AMS, 124(10) , 1 9 96, pp. 2993 - 3002. [2] T. M. Cov er and J. A. Thomas. El e ments of Information The ory , John Wiley & Sons, Inc., N. Y. , 1991. 7 [3] G. Kalai, The entr opy/influenc e c onje ctur e , http://terrytao.w ordp ress.com/2007/08/1 6/gil-kalai-the-entrop yinfluence-conjecture/ [4] N. K eller, E. Mossel, and T. Sc hla nk, A Note on the En tr opy/In fluenc e C onje ctur e , a rXiv :1 105.2651v1 , 2011 [5] A. Kliv ans, H. Lee, and A. W an. Manso ur’s Conje ctur e i s true for r andom DNF fo rm u- las , COL T , 2 010, pp. 368- 380. [6] Y. Mansour, L e arning B o ole an functions via the F ourier tr ansform , Kluw er Academic Publishers , 1994, pp. 391- 424. [7] R. O’donnell, J. W right and Y. Zhou, The F ourier Entr opy-Influenc e Conje ctur e for c ertain classes o f Bo ole an functions , ICALP , 2011, pp. 33 0-341. 8