Fourier Analysis of Boolean Functions Juntas, Projections, Influences Etc.

© S.Safra Boolean Functions and Juntas A boolean function Def: f is a j-Junta if there exists J  [n] where |J|≤ j, and s.t. for every x f(x) = f(x  J) f is ( , j)-Junta if  j-Junta f’ s.t. f is ( , j)-Junta if  j-Junta f’ s.t.

© S.Safra 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. We would tend to omit p

© S.Safra Background Thm (Friedgut): a boolean function f with small average-sensitivity is an [ ,j]-junta 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 (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 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 Corollary: a boolean function f with small noise- sensitivity is an [ ,j]-junta Parameters: average-sensitivity, high-frequency weight, noise-sensitivity Parameters: average-sensitivity, high-frequency weight, noise-sensitivity

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

© S.Safra Noise-Sensitivity Def( ,p,x [n] ): Let 0< <1, and x  P([n]). Then y~ ,p,x, if y = (x\I)  z where 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 I~  [n] is a noise subset, and z~  p I is a replacement. z~  p I 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. 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

© S.Safra Noise-Sensitivity – Cont. Advantage: very efficiently testable (using only two queries) by a perturbation-test. 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. 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 Prop: the -noise-sensitivity is given by

© S.Safra f f * * 0* 1* 11* 110* 00* 01* 010* 011* 000* 001* 111* 10* 100* 101* Functions as an Inner-Product Vector-Space f f 2n2n 2n2n * * 0* 1* 11* 110* 00* 01* 010* 011* 000* 001* 111* 10* 100* 101*

© S.Safra Functions as an Inner-Product Vector-Space A functions f is a vector A functions f is a vector Inner product (normalized) Inner product (normalized) Norm (normalized) Norm (normalized)

© S.Safra Simple Observations Claims: For a boolean f

© S.Safra Fourier-Walsh Transform Consider all multiplicative functions, one for each character S  [n] Consider all multiplicative functions, one for each character S  [n] Given any function let the Fourier-Walsh coefficients of f be Given any function let the Fourier-Walsh coefficients of f be thus f can described as thus f can described as

© S.Safra Fourier Transform: Norm Norm: (not normalized) Thm [Parseval]: Hence, for a boolean f

© S.Safra Simple Observations Claim: Claim: Hence, for any f Hence, for any f

Putting a Junta to the Test Joint work with Eldar Fischer & Guy Kindler Building on [KKL,Freidgut,Bourgain]

© S.Safra Junta Test Def: A Junta test is as follows: A distribution over l queries For each l-tuple, a local-test that either accepts or rejects:T[x 1, …, x l ]: {1, -1} l  {T,F} s.t. for a j-junta f whereas for any f which is not ( , j)-Junta

© S.Safra Variables` Influence The influence of an index i  [n] on a boolean function f:{1,-1} n  {1,-1} is The influence of an index i  [n] on a boolean function f:{1,-1} n  {1,-1} is Which can be expressed in terms of the Fourier coefficients of f Claim: Which can be expressed in terms of the Fourier coefficients of f Claim:

© S.Safra Fourier Representation of influence Proof: consider the I-average function on P[I] which in Fourier representation is and

© S.Safra Fourier Representation of influence Proof: consider the influence function which in Fourier representation is and

© S.Safra High vs Low Frequencies Def: The section of a function f above k is and the low-frequency portion is

© S.Safra Subsets` Influence Def: The influence of a subset I  [n] on a boolean function f is and the low-frequency influence

© S.Safra Independence-Test The I-independence-test on a boolean function f is, for Lemma:

© S.Safra

Junta Test The junta-size test on a boolean function f is The junta-size test on a boolean function f is Randomly partition [n] to I 1,.., I r Randomly partition [n] to I 1,.., I r Run the independence-test t times on each I h Run the independence-test t times on each I h Accept if ≤j of the I h fail their independence-tests Accept if ≤j of the I h fail their independence-tests For r>>j 2 and t>>j 2 / 

© S.Safra Completeness Lemma: for a j-junta f Proof: only those sets which contain an index of the Junta would fail the independence-test

© S.Safra Soundness Lemma: Proof: Assume the premise. Fix  <<1/t and let

© S.Safra |J| ≤ j Prop: r >> j implies |J| ≤ j Proof: otherwise, J spreads among I h w.h.p. J spreads among I h w.h.p. and for any I h s.t. I h  J ≠  it must be that influence I (f) >  and for any I h s.t. I h  J ≠  it must be that influence I (f) > 

© S.Safra High Frequencies Contribute Little Prop: k >> r log r implies Proof: a character S of size larger than k spreads w.h.p. over all parts I h, hence contributes to the influence of all parts. If such characters were heavy (>  /4), then surely there would be more than j parts I h that fail the t independence-tests

© S.Safra Almost all Weight is on J Lemma: Proof: otherwise, since for a random partition w.h.p. (Chernoff bound) for every h however, since for any I the influence of every I h would be ≥  /100rk

© S.Safra Find the Close Junta Now, since consider the (non boolean) which, if rounded outside J is boolean and not more than  far from f

© S.Safra Open Problems Is there a characterization, via Fourier transform, of all efficiently testable properties? Is there a characterization, via Fourier transform, of all efficiently testable properties? What about tests that probe f only at two or three points? With applications to hardness of approximation. What about tests that probe f only at two or three points? With applications to hardness of approximation.

© S.Safra Consider the q-biased product distribution  q : Def: The probability of a subset F and for a family of subsets  Consider the q-biased product distribution  q : Def: The probability of a subset F and for a family of subsets  Product, Biased Distribution

© S.Safra Beckner/Nelson/Bonami Inequality Def: let T  be the following operator on any f, Prop: Proof:

© S.Safra 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

© S.Safra Beckner/Nelson/Bonami Corollary Corollary: for f s.t. f >k =0 and p≥r≥1 Proof:

© S.Safra Average Sensitivity The sum of variables’ influence is referred to as the average sensitivity Which can be expressed by the Fourier coefficients as

© S.Safra Freidgut Theorem Thm: any boolean f is an [ , j]-junta for Proof: 1. Specify the junta J 2. Show the complement of J has little influence

© S.Safra Specify the Junta Set k=O(as(f)/  ), and  =2 -O(k) Let We’ll prove: and let hence, J is a [ ,j]-junta, and |J|=2 O(k)

© S.Safra High Frequencies Contribute Little Prop: Proof: a character S of size larger than k contributes k times the square of its coefficient to the average sensitivity. If such characters were heavy (>  /4), as(f) would have been large

© S.Safra Altogether Lemma: Proof:

© S.Safra Altogether

Biased  q - Influence The  q -influence of an index i  [n] on a boolean function f:P[n]  {1,-1} is The  q -influence of an index i  [n] on a boolean function f:P[n]  {1,-1} is

© S.Safra Thm [Margulis-Russo]: For monotone  Hence Lemma: For monotone   > 0,  q  [p, p+  ] s.t. as q (  )  1/  Proof: Otherwise  p+  (  ) > 1

© S.Safra Proof [Margulis-Russo]:

© S.Safra Erdös-Ko-Rado Def: A family of subsets   P[R] is t-intersecting if for every F 1, F 2  , |F 1  F 2 |  t Def: A family of subsets   P[R] is t-intersecting if for every F 1, F 2  , |F 1  F 2 |  t Thm[Wilson,Frankl,Ahlswede-Khachatrian]: For a t-intersecting , where Thm[Wilson,Frankl,Ahlswede-Khachatrian]: For a t-intersecting , where Corollary:  p (  ) > P    is not 2-intersecting Corollary:  p (  ) > P    is not 2-intersecting P  = P  =