Noise-Insensitive Boolean-Functions are Juntas Guy Kindler & Muli Safra

Dictatorship Def: a boolean function P([n]){-1,1} is a monotone e-dictatorships --denoted fe--if:

We would tend to omit p Juntas Def: a boolean function f:P([n]){-1,1} is a j-Junta if J[n] where |J|≤ j, s.t. for every x[n]: f(x) = f(x  J) Def: f is an [, j]-Junta if  j-Junta f’ s.t. Def: f is an [, j, p]-Junta if  j-Junta f’ s.t.

Codes and Boolean Functions Def: a code is a mapping of a set of n elements (log n bits’ string) to a set of m-bits strings C:[n]{0,1}m, i.e. C(e) = a1…am Def: Let Sj={e[n] | C(e)j=T} Let ={Sj}j[m] C(1) C(2) C(3) C(n) FF T TF F TT F TT T S1={2,3,n} Sm={1,n}

Codes and Boolean Functions Def: Let Ee be the encoding of element e. Consider {Ee}e[n] Each Ee’s truth-table represents a legal--code-word of C ( since C(e) = Ee(S1)…Ee(Sm) ) C(1) C(2) C(3) C(n) FF0 T TF F TT F TT T S1={2,3,n} Sm={1,n}

Long-Code In the long-code L:[n] {0,1}2n each element is encoded by an 2n-bits This is the most extensive code, as  = P([n]), i.e. the bits represent all subsets in P([n])

Long-Code Encoding an element e[n]: Ee legally-encodes an element e if Ee = fe F F T T T

Motivation – Testing Long-code Def (a long-code test): given a code-word w, probe it in a constant number of entries, and accept w.h.p if w is a monotone dictatorship reject w.h.p if w is not close to any monotone dictatorship

Motivation – Testing Long-code Def(a long-code list-test): given a code-word w, probe it in a constant number of entries, and accept w.h.p if w is a monotone dictatorship, reject w.h.p if w is not even approximately determined by a small list of domain elements, that is, if  a Junta J[n] s.t. f is close to f’ and f’(x)=f’(xJ) for all x Note: a long-code list-test, distinguishes between the case w is a dictatorship, to the case w is far from a junta.

Motivation – Testing Long-code The long-code test, and the long-code list-test are essential tools in proving hardness results. Examples … Hence finding simple sufficient-conditions for a function to be a junta is important.

Background Thm (Friedgut): a boolean function f with small average-sensitivity is an [,j]-junta Thm (Bourgain): a boolean function f with small high-frequency weight is an [,j]-junta Thm (Kindler&Safra): a boolean function f with small high-frequency weight in a p-biased measure is an [,j]-junta Corollary: a boolean function f with small noise-sensitivity is an [,j]-junta Parameters: average-sensitivity, high-frequency weight, noise-sensitivity

Noise-Sensitivity Idea: check how the value of f changes when the input is changed not on one, but on several coordinates. [n] I z x

Noise-Sensitivity Def(,p,x[n] ): Let 0<<1, and xP([n]). Then y~,p,x, if y = (x\I) z where I~[n] is a noise subset, and z~ pI is a replacement. Def(-noise-sensitivity): let 0<<1, then Note: deletes a coordinate in x w.p. (1-p), adds a coordinate to x w.p. p. Hence, when p=1/2: equivalent to flipping each coordinate in x w.p. /2. [n] x I z

Noise-Sensitivity – Cont. Advantage: very efficiently testable (using only two queries) by a perturbation-test. Def (perturbation-test): choose x~p, and y~,p,x, check whether f(x)=f(y). The success is proportional to the noise-sensitivity of f. Prop: the -noise-sensitivity is given by

Relation between Parameters Prop: small ns small high-freq weight Proof: therefore: if ns is small, then Hence the high frequencies must have small weights (as ). Prop: small as small high-freq weight Proof:

Average and Restriction Def: Let I[n], xP([n]\I), the restriction function is Def: the average function is Note: I y x [n] I y y y y y x

Fourier Expansion Prop: Corollary:

Variation Def: the variation of f (formerly called influence): Prop: the following are equivalent definitions to the variation of f:

Proof Recall Therefore

Proof – Cont. Recall Therefore (by Parseval):

High/Low Frequencies and their Weights Def: the high-frequency portion of f: Def: the low-frequency portion of f: Def: the high-frequency-weight is: Def: the low-frequency-weight is:

Low-freq variation and Low-freq average-sensitivity Def: the low-frequency variation is: Def: the average sensitivity is And in Fourier representation: Def: the low-frequency average sensitivity is:

Main Results Theorem:  constant >0 s.t. any boolean function f:P([n]){-1,1} satisfying is an [,j]-junta for j=O(-2k32k). Corollary: fix a p-biased distribution p over P([n]). Let >0 be any parameter. Set k=log1-(1/2). Then  constant >0 s.t. any boolean function f:P([n]){-1,1} satisfying is an [,j]-junta for j=O(-2k32k).

First Attempt: Following Freidgut’s Proof Thm: any boolean function f is an [,j]-junta for Proof: Specify the junta where, let k=O(as(f)/) and fix =2-O(k) Show the complement of J has small variation P([n]) J

Following Freidgut - Cont P([n]) J Following Freidgut - Cont Lemma: Proof: Now, lets bound each argument: Prop: Proof: characters of sizek contribute to the average-sensitivity at least (since )

Following Freidgut - Cont True only since this is a {-1,0,1} function. So we cannot proceed this way with only ask!  we do not know whether as(f) is small!  Following Freidgut - Cont Prop: Proof:

Important Lemma Lemma: >0, s.t. for any  and any function g:P([m]) , the following holds: Low-freq high-freq

Beckner/Nelson/Bonami Inequality Def: let T be the following operator on any f, Thm: for any p≥r and ≤((r-1)/(p-1))½ Corollary: for f s.t. f>k=0

Probability Concentration Simple Bound: Proof: Low-freq Bound: Let g:P([m])  s.t. g>k=0, let >0, then >0 s.t. Proof: recall the corollary: 

Lemma’s Proof Now, let’s prove the lemma: Bounding low and high freq separately: , Low-freq bound simple bound

Shallow Function Def: a function f is linear, if only singletons have non-zero weight Def: a function f is shallow, if f is either a constant or a dictatorship. Claim: boolean linear functions are shallow. weight Character size 0 1 2 3 k n

Boolean Linear  Shallow Claim: boolean linear functions are shallow. Proof: let f be boolean linear function, we next show: {io} s.t. (i.e. ) And conclude, that either or i.e. f is shallow

Claim 1 Claim 1: let f be boolean linear function, then {io} s.t. Proof: w.l.o.g assume for any z{3,…,n}, consider x00=z, x10=z{1}, x01=z{2}, x11=z{1,2} then . Next value must be far from {-1,1}, A contradiction! (boolean function) Therefore 1 -1 ?

Claim 2 Claim 2: let f be boolean function, s.t. Then either or Proof: consider f() and f(i0): Then but f is boolean, hence therefore 1 -1

Proving FKN: almost-linear  close to shallow Theorem: Let f:P([n])  be linear, Let let i0 be the index s.t. is maximal then Note: f is linear, hence w.l.o.g., assume i0=1, then all we need to show is: We show that in the following claim and lemma.

Corollary Corollary: Let f be linear, and then  a shallow boolean function g s.t. Proof: let , let g be the boolean function closest to l. Then, this is true, as is small (by theorem), and additionally is small, since

Each of weight no more than c Claim 1 Claim 1: Let f be linear. w.l.o.g., assume then global constant c=min{p,1-p} s.t. {} {1} {2} {i} {n} {1,2} {1,3} {n-1,n} S {1,..,n} weight Characters Each of weight no more than c

Proof of Claim1 Proof: assume for any z{3,…,n}, consider x00=z, x10=z{1}, x01=z{2}, x11=z{1,2} then Next value must be far from {-1,1} ! A contradiction! (to ) 1 -1 ?

Lemma Lemma: Let g be linear, let assume , then note Lemma: Let g be linear, let assume , then Corrolary: The theorem follows from the combination of claim1 and the lemma: Let m be the minimal index s.t. Consider If m=2: the theorem is obtained (by lemma) Otherwise -- a contradiction to minimality of m :

Lemma’s Proof Lemma’s Proof: Note Hence, all we need to show is that Intuition: Note that |g| and |b| are far from 0 (since |g| is -close to 1, and c-close to b). Assume b>0, then for almost all inputs x, g(x)=|g(x)| (as ) Hence E[g]  E[|g(x)|], and therefore var(g)  var(|g|)

E2[g] - E2[|g|] = 2E2[|g|1{f<0}]  o() (by Azuma’s inequality) Proof-map: |g|,|b| are far from 0 g(x)=|g(x)| for almost all x E[g]  E[|g|] var(g)  var(|g|)    E2[g] - E2[|g|] = 2E2[|g|1{f<0}]  o() (by Azuma’s inequality) We next show var(g)  var(|g|): By the premise however therefore 

Central Ideas: Linear Functions and Random Partition Idea 1: recall (theorem’s premise) Assume f is close to linear then f is close to shallow (by [FKN]). Idea 2: Let . Partition J into I1,…,Ir. r is large, hence w.h.p fI[x] is close to linear (low freq characters intersect each I by 1 element). P([n]) I2 Ir I I1 J

Variation Lemma Lemma(variation): >0, and r>>k s.t. P([n]) Variation Lemma I2 Ir I I1 Lemma(variation): >0, and r>>k s.t. Corollary: for most I and x, fI[x] is almost constant J

Using Idea2 By union bound on I1,…,Ir: P([n]) Using Idea2 I2 Ir I I1 By union bound on I1,…,Ir: (set ) Let f’(x) = sign( AJ[f](xJ) ). f’ is the boolean function closest to AJ[f], therefore Hence f is an [,j]-junta. J

variation-Lemma - Proof Plan Lemma(variation): >0, and r>>k s.t. Sketch for proving the variation lemma: w.h.p fI[x] is almost linear w.h.p fI[x] is close to shallow fI[x] cannot be close to dictatorship too often.

Proof-map: w.h.p fI[x] is almost linear w.h.p fI[x] is close to shallow fI[x] cannot be close to dictatorship too often. Lemma Proof The low frequencies characters are small, r is rather large, hence w.h.p the characters are linear at each I. P([n]) I2 Ir I I1 J

almost linear  almost shallow Proof-map: w.h.p fI[x] is almost linear w.h.p fI[x] is close to shallow fI[x] cannot be close to dictatorship too often. almost linear  almost shallow Theorem([FKN]): global constant M, s.t. boolean function f, shallow boolean function g, s.t. Hence, ||fI[x]>1||2 is small  fI[x] is close to shallow!

Preliminary Lemma and Props Prop: if fI[x] is a dictatorship, then coordinate i s.t. (where p is the bias). Corollary (from [FKN]): global constant M, s.t. boolean function h, either or weight Total weight of no more than 1-p Characters {1} {2} {i} {n} {1,2} {1,3} {n-1,n} S {1,..,n}

Proof-map: w.h.p fI[x] is almost linear w.h.p fI[x] is close to shallow fI[x] cannot be close to dictatorship too often. Few Dictatorships Lemma: >0, s.t. for any  and any function g:P([m]) , the following holds: Def: Let DI be the set of xP(I), s.t. fI[x] is a dictatorship, i.e. Next we show, that |DI| must be small, hence for most x, fI[x] is constant.

|DI| must be small Lemma: Proof: let , then Prev lemma Parseval Each S is counted only for one index iI. (Otherwise, if S was counted for both i and j in I, then |SI|>1!)

Simple Prop Prop: let {ai}iI be sub-distribution, that is, iIai1, 0ai, then iIai2maxiI{ai}. Proof: 1 2 3 max n ai no more than 1 1 1 2 3 n ai 1/amax 1

|DI| must be small - Cont Therefore (since ), Hence

Obtaining the Lemma It remains to show that indeed: Prop1: Prop2: Recall However {S}S are orthonormal, and

Obtaining the Lemma – Cont. Prop3: Proof: separate by freq: Small freq: Large freq: Corollary(from props 2,3):

Obtaining the Lemma – Cont. Recall: by corollary from [FKN], Either or Hence By Corollary Combined with Prop1 we obtain: |DI| is small

The End

XOR Test Let  be a random procedure for choosing two disjoint subsets x,y s.t.: i[n], ix\y w.p 1/3, iy\x w.p 1/3, and ixy w.p 1/3. Def(XOR-Test): Pick <x,y>~, Accept if f(x)f(y), Reject otherwise.

Example Claim: Let f be a dictatorship, then f passes the XOR-test w.p. 2/3. Proof: Let i be the dictator, then Pr<x,y>~[f(x)f(y)]=Pr<x,y>~ [ixy]=2/3 Claim: Let f’ be a -close to a dictatorship f, then f’ passes the XOR-test w.p. 2/3 – 2/3(-2). Proof: see next slide…

Local Maximality Def: f is locally maximal with respect to a test, if f’ obtained from f by a change on one input x0, that is, Pr<x,y>~[f(x)f(y)]  Pr<x,y>~[f’(x)f’(y)] Def: Let x be the distribution of all y such that <x,y>~. Claim: if f is locally maximal then f(x) = -sign(Ey~(x)[f(y)]).

Claim: Proof: immediate from the Fourier-expansion, and the fact that yx=

Conjecture: Let f be locally maximal (with respect to the XOR-test), assume f passes the XOR-test w.p  1/2 + , for some constant >0, then f is close to a junta.