Fourier Analysis,Fourier Analysis,Projections,Projections, Influence, Influence,

Junta,Junta,Etc…Etc…

Fourier Analysis,Fourier Analysis,Projections,Projections, Influence, Influence,

Junta,Junta,Etc…Etc…

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Boolean Functions and Boolean Functions and JuntasJuntas

DefDef: A : A Boolean functionBoolean function

n

f : P n 1,1

P n x n

1,1

n

f : P n 1,1

P n x n

1,1

©©S.SafraS.Safra

ff**

-1*-1*

1*1*

11*11*

11-1*

11-1*

-1-1*-1-1*

-11*-11*

-11-1*-11-1*

-111*

-111*

-1-1-1*-1-1-1*

-1-11*-1-11*

111*111*

1-1*1-1*1-1-1*

1-1-1*

1-11*

1-11*

Functions as anFunctions as anInner-Product Vector-SpaceInner-Product Vector-Space

ff2n2n*

*

-1*-1*

1*1*

11*11*

11-1*11-1*

-1-1*-1-1*

-11*-11*

-11-1*-11-1*

-111*-111*

-1-1-1*-1-1-1*

-1-11*-1-11*

111*111*

1-1*1-1*

1-1-1*1-1-1*

1-11*1-11*

©©S.SafraS.Safra

Functions’ Vector-Space Functions’ Vector-Space A functions A functions ff is a vector is a vector

Addition:Addition:

‘f+g’(x) = f(x) + g(x)‘f+g’(x) = f(x) + g(x)

Multiplication by scalarMultiplication by scalar

‘c‘cf’(x) = cf’(x) = cf(x)f(x)

n2f n2f

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

©©S.SafraS.Safra

Variables` InfluenceVariables` Influence

The The influenceinfluence of an index of an index i i [n][n] on a on a Boolean function Boolean function f:{1,-1}f:{1,-1}nn {1,-1}{1,-1} is is

x P n(f ) Pr f x f x i

iInfluence

x P n(f ) Pr f x f x i

iInfluence

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Norms Norms DefDef: : ExpectationExpectation Norm Norm

DefDef: : SumSum Norm Norm

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

q

qq

x P n

ff xE

q

qq

x P n

ff xE

qq

qx P n

ff x

qq

qx P n

ff x

©©S.SafraS.Safra

Inner-ProductInner-Product

A functions A functions ff is a vector is a vector

Inner product (normalized)Inner product (normalized)

n2f n2f

nx 2

f g f x g xE

nx 2

f g f x g xE

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

©©S.SafraS.Safra

Simple ObservationsSimple Observations

ClaimsClaims::

For any function For any function ff whose range is whose range is {-1, 0, 1}{-1, 0, 1}

1 xf E f(x) 1 xf E f(x)

p 1

p 1 xff Pr f x 1,1 p 1

p 1 xff Pr f x 1,1

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

©©S.SafraS.Safra

MonomialsMonomials

What would be the monomials over What would be the monomials over x x P[n]P[n]??

All powers except All powers except 00 and and 11 disappear! disappear!

Hence, one for each Hence, one for each charactercharacter SS[n][n]

These are all the multiplicative functionsThese are all the multiplicative functions

S xS i

i S

(x) x 1

S xS i

i S

(x) x 1

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

©©S.SafraS.Safra

Fourier-Walsh TransformFourier-Walsh Transform

Consider all charactersConsider all characters

Given any functionGiven any functionlet the let the Fourier-Walsh coefficientsFourier-Walsh coefficients of of ff be be

thus thus ff can described as can described as

f : P n f : P n

S ii S

(x) x

S ii S

(x) x

Sf S f Sf S f

SS

ff S SS

ff S

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

©©S.SafraS.Safra

Fourier Transform: NormFourier Transform: Norm

NormNorm: (: (SumSum))

ThmThm [Parseval]: [Parseval]:

Hence, for a Boolean Hence, for a Boolean ff

q q

q S n

ff S

q q

q S n

ff S

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

ff**

0*0*

1*1*

11*11*

110*110*

00*00*

01*01*

010*010*

011*011*

000*000*

001*001*

111*111*

10*10*

100*100*

101*101*

22ff

22ff

2 2

2S

f (S) f 1 2 2

2S

f (S) f 1

©©S.SafraS.Safra

Variables` InfluenceVariables` Influence

The The influenceinfluence of an index of an index i i [n][n] on a on a Boolean function Boolean function f:{1,-1}f:{1,-1}nn {1,-1}{1,-1} is is

Which can be expressed in terms of the Which can be expressed in terms of the Fourier coefficients of Fourier coefficients of ff

ClaimClaim::

x P n(f ) Pr f x f x i

iInfluence

x P n(f ) Pr f x f x i

iInfluence

2

S,i S

ff S

iInfluence

2

S,i S

ff S

iInfluence

©©S.SafraS.Safra

Restriction and AverageRestriction and Average

DefDef: Let : Let II[n], x[n], xP([n]\I),P([n]\I), the the restriction function restriction function isis

DefDef: the : the average function average function isis

NoteNote::

I

Iy P I

A f : P I

A f x E f x y

I

Iy P I

A f : P I

A f x E f x y

I

I

f x : P I 1,1

f x y f x y

I

I

f x : P I 1,1

f x y f x y

I Iy P I

A f x E f x y

I Iy P I

A f x E f x y

[n]I

x

y

[n]I

x

y y

y yy

©©S.SafraS.Safra

In Fourier ExpansionIn Fourier Expansion

PropProp: :

And since the expectation of a function is And since the expectation of a function is its coefficient on the empty character:its coefficient on the empty character:

CorollaryCorollary::

CorollaryCorollary::

I SS I

A ff (S)

I S

S I

A ff (S)

I STS I T I S

f x f T x

I STS I T I S

f x f T x

2 2

i 2S,i S

f 1 A ff SiInfluence

2 2

i 2S,i S

f 1 A ff SiInfluence

©©S.SafraS.Safra

Expectation and VarianceExpectation and Variance

ClaimClaim::

Hence, for any Hence, for any ff

x

f E f(x)

xf E f(x)

22

x P n x P n

2 22

2S n,S

ff x E f x

ff f S

V E

22

x P n x P n

2 22

2S n,S

ff x E f x

ff f S

V E

©©S.SafraS.Safra

Average SensitivityAverage Sensitivity

DefDef: the sensitivity of : the sensitivity of xx w.r.t. w.r.t. ff is is

DefDef: the average-sensitivity of : the average-sensitivity of ff is is

i

# f x f x i i

# f x f x i

ix

ii

2

S

as f E # f x f x i

f

f S S

Influence

ix

ii

2

S

as f E # f x f x i

f

f S S

Influence

©©S.SafraS.Safra

When When as(f)=1as(f)=1

DefDef: a balanced function : a balanced function ff is s.t. is s.t.

ThmThm: a balanced, Boolean : a balanced, Boolean ff s.t. s.t. as(f)=1as(f)=1 is is a dictatorshipa dictatorship

ProofProof: observe that: observe that

and since and since as(f)=1 as(f)=1 it must be thatit must be that

however however ||f||||f||22=1=1 hence hence

xE f(x) 0 xE f(x) 0

ii

ffi ii

ffi

f 0 f 0

2

S n,S

f S S 1

2

S n,S

f S S 1

©©S.SafraS.Safra

Linear, Boolean FunctionsLinear, Boolean Functions

Proof(cont)Proof(cont)::

pick any pick any xx; ; f(x) f(x) {-1, 1} {-1, 1}

Pick Pick {i}{i} with non-zero coefficient with non-zero coefficient

Observe that Observe that f(x f(x {i}) {i}) {-1, 1} {-1, 1} however however differ from differ from f(x)f(x)

Conclusion: Conclusion: f ( i ) 1,1 f ( i ) 1,1

ii

ffi x ii

ffi x

©©S.SafraS.Safra

Codes and Boolean Codes and Boolean FunctionsFunctions

DefDef: an : an mm-bit code is a subset of the set of all the -bit code is a subset of the set of all the mm-binary string -binary string

CC{-1,1}{-1,1}mm

The distance of a code The distance of a code CC, which is the minimum, , which is the minimum, over all pairs of legal-words (in over all pairs of legal-words (in CC), of the ), of the Hamming distance between the two wordsHamming distance between the two words

A Boolean function over A Boolean function over nn binary variables, binary variables,is a is a 22nn-bit string-bit string

Hence, a set of Boolean functions can be Hence, a set of Boolean functions can be considered as a considered as a 22nn-bits code-bits code

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Hadamard CodeHadamard Code

In the Hadamard code the set In the Hadamard code the set of legal-words consists of all of legal-words consists of all multiplicative (linear if over multiplicative (linear if over {0,1}{0,1}) functions) functions

C={C={SS | S | S [n]} [n]}

namely all characters namely all characters

2222

Hadamard Test – SoundnessHadamard Test – Soundness

PropProp(soundness):(soundness):

ProofProof::

1 2 3

1 2 3

1 2 3 3

1 2 3

1 3 2 3

1 2 3

x,y

1 2 3 x,y S S SS ,S ,S

1 2 3 x,y S S S SS ,S ,S

1 2 3 x S S y S SS ,S ,S

3

S

<E [f (x) f (y) f(xy)]=

= f S f S f S E [ (x) (y) (xy)]=

= f S f S f S E [ (x) (y) (x) (y)]=

= f S f S f S E [ (x) (x)] E [ (y) (y)]=

= f S

1+Pr[f (x) f (y) f(xy)]> S [n],f S

2

©©S.SafraS.Safra

Long-CodeLong-Code

In the long-code the set of legal-words consists of all In the long-code the set of legal-words consists of all monotone dictatorshipsmonotone dictatorships

This is the most extensive binary code, as its bits This is the most extensive binary code, as its bits represent all possible binary values over represent all possible binary values over nn elements elements

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Long-CodeLong-Code

Encoding an element Encoding an element ee[n][n] :: EEee legally-encodeslegally-encodes an element an element ee if if EEee = f = fee

FF FF TT TT TT

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Testing Long-codeTesting Long-code

DefDef(a (a long-code list-testlong-code list-test): given a code-word ): given a code-word ff, , probe it in a constant number of entries, andprobe it in a constant number of entries, and accept almost always if accept almost always if ff is a monotone is a monotone

dictatorshipdictatorship reject w.h.p if reject w.h.p if ff does not havedoes not have a sizeable fraction a sizeable fraction

of its Fourier weight concentrated on a small set of its Fourier weight concentrated on a small set of variables, that is, if of variables, that is, if a a semi-Juntasemi-Junta JJ[n][n] s.t. s.t.

NoteNote: a long-code list-test, distinguishes : a long-code list-test, distinguishes between the case between the case ff is a is a dictatorshipdictatorship, to the , to the case case ff is far from a is far from a juntajunta..

2

S J

f S

2

S J

f S

©©S.SafraS.Safra

Motivation – Testing Long-codeMotivation – Testing Long-code

TheThe long-code list-test long-code list-test are essential tools are essential tools in proving hardness results. in proving hardness results.

Hence finding simple sufficient-conditions Hence finding simple sufficient-conditions for a function to be a junta is important.for a function to be a junta is important.

©©S.SafraS.Safra

Perturbation Perturbation

DefDef: denote by : denote by the distribution the distribution over all subsets of over all subsets of [n][n], which , which assigns probability to a subset assigns probability to a subset xx as follows:as follows:

independently, for each independently, for each ii[n][n], let, let iixx with probability with probability 1-1- iixx with probability with probability

©©S.SafraS.Safra

Long-Code TestLong-Code Test

Given a Boolean Given a Boolean ff, choose , choose random random xx and and yy, and choose , and choose zz; check that; check that

f(x)f(y)=f(xyz)f(x)f(y)=f(xyz)

PropProp(completeness): a legal (completeness): a legal long-code word (a dictatorship) long-code word (a dictatorship) passes this test w.p. passes this test w.p. 1-1-

2929

Long-code Test – SoundnessLong-code Test – Soundness

PropProp(soundness):(soundness):

ProofProof::

1 2 3

1 2 3

1 2 3 3 3

1 2 3

1 3 2 3

1 2 3

x,y

1 2 3 x,y,z S S SS ,S ,S

1 2 3 x,y,z S S S S SS ,S ,S

1 2 3 x S S y S SS ,S ,S

<E [f (x) f (y) f(xyz)]=

= f S f S f S E [ (x) (y) (xyz)]=

= f S f S f S E [ (x) (y) (x) (y) (z)]=

= f S f S f S E [ (x) (x)] E [ (y) (

3z S

S3

S

y)] E [ (z)]=

= f S 1 2

S J

1+Pr[f (x) f (y) f(xyz)]> J [n], f S 22

©©S.SafraS.Safra

VariationVariation

DefDef: the : the variationvariation of of ff (extension of (extension of influenceinfluence))

PropProp: the following are equivalent : the following are equivalent definitions to the definitions to the variationvariation of of ff::

22

I I 2S I

ff A ff S

variation 22

I I 2S I

ff A ff S

variation

I Iy P Ix P I

f E var f x y

variation

I I

y P Ix P If E var f x y

variation

©©S.SafraS.Safra

ProofProof

RecallRecall

ThereforeTherefore

2

I Iy P I

S I ,S

var f x y f x S

2

I Iy P I

S I ,S

var f x y f x S

I TT I S

f x S f T x

I TT I S

f x S f T x

2 2

Iy P Ix P I T: , T IT I S

S

E var f x y f T f T

2 2

Iy P Ix P I T: , T IT I S

S

E var f x y f T f T

2 2

IT I S

E f x S f T

2 2

IT I S

E f x S f T

©©S.SafraS.Safra

Proof – Cont.Proof – Cont.

RecallRecall

Therefore (by Parseval):Therefore (by Parseval):

I SS I

A ff (S)

I S

S I

A ff (S)

2 22

I 2S I S I

2

S I

f A ff S f S f S 0

f S

2 22

I 2S I S I

2

S I

f A ff S f S f S 0

f S

©©S.SafraS.Safra

High vs Low FrequenciesHigh vs Low Frequencies

DefDef: The section of a function : The section of a function ff above above kk is is

and the and the low-frequency low-frequency portion isportion is

kS

S k

ff S

k

SS k

ff S

kS

S k

ff S

k

SS k

ff S

©©S.SafraS.Safra

Junta TestJunta Test

DefDef: A : A JuntaJunta testtest is as follows: is as follows:A distribution over A distribution over ll queries queries

For each For each ll-tuple, a local-test that either accepts or -tuple, a local-test that either accepts or rejects:rejects: T[xT[x11, …, x, …, xll]: {1, -1}]: {1, -1}ll{T,F}{T,F}

s.t. for a s.t. for a jj-junta -junta ff

whereas for any whereas for any ff which is not which is not ((, j)-, j)-JuntaJunta

l: P n 0,1 l: P n 0,1

1 lx ,..,x 1 lPr T x ,..,x f 1 1 lx ,..,x 1 lPr T x ,..,x f 1

1 lx ,..,x 1 l

1Pr T x ,..,x (f ) 2 1 lx ,..,x 1 l

1Pr T x ,..,x (f ) 2

©©S.SafraS.Safra

Fourier Representation of Fourier Representation of influenceinfluence

ProofProof: consider the : consider the II-average function on -average function on P[P[II]]

which in Fourier representation iswhich in Fourier representation is

andand

I

y P IA f (x) E f x y

I

y P IA f (x) E f x y

I SS I

A ff (S)

I S

S I

A ff (S)

2 2

i i 2i S

f 1 A ff (S)

influence

2 2

i i 2i S

f 1 A ff (S)

influence

©©S.SafraS.Safra

Fourier Representation of Fourier Representation of influenceinfluence

ProofProof: consider the influence : consider the influence functionfunction

which in Fourier representation iswhich in Fourier representation is

andand

i

f x f x if x

2

i

f x f x if x

2

i S S SS S

Si S

1 1f x f(S) x f(S) x i

2 2

f(S) x

i S S SS S

Si S

1 1f x f(S) x f(S) x i

2 2

f(S) x

22

i i 2i S

ff f (S)

influence 22

i i 2i S

ff f (S)

influence

©©S.SafraS.Safra

Subsets` InfluenceSubsets` Influence

DefDef: The : The VariationVariation of a subset of a subset I I [n] [n] on a on a Boolean function Boolean function ff is is

and the and the low-frequency influencelow-frequency influence

2 2

I I2 S I

f 1 A ff S

Variation 2 2

I I2 S I

f 1 A ff S

Variation

2

k kI I

S IS k

ff f S

Variation Variation 2

k kI I

S IS k

ff f S

Variation Variation

©©S.SafraS.Safra

Independence-TestIndependence-Test

The The II-independence-test-independence-test on a Boolean on a Boolean function function ff is, for is, for

LemmaLemma::

?

1 2

1 2 1 2

w I , z ,z I

I T(w, z ,z ) f w z f w z:

?

1 2

1 2 1 2

w I , z ,z I

I T(w, z ,z ) f w z f w z:

1 2

11 2 I2

w P Iz ,z P I

Pr I T(w, z ,z ) 1 f

Variation

1 2

11 2 I2

w P Iz ,z P I

Pr I T(w, z ,z ) 1 f

Variation

©©S.SafraS.Safra

I I

x P Iy ,y P I1 2

2I

2 21 A f x 1 A f x

1 2 2 2x P[n

22 2 A f x 1I24 2x P[n]

1I

]

2

Pr I T(x, y

E 1 1 A f

,y E

1 f

)

influence

I I

x P Iy ,y P I1 2

2I

2 21 A f x 1 A f x

1 2 2 2x P[n

21

I2 2

]

2 2 A f x

4x P[n]

1I2

Pr I T(x, y

1 1 A f

y E

f

,

1

)

E

influence

I I

x P Iy ,y P I1 2

2I

2 21 A f x 1 A f x

1 2 2 2x P[n]

22 2 A f x 1I24

1I

2x P[n]

2

Pr I T(x, y ,y ) E

E 1 1 A f

1 f

influence

I I

x P Iy ,y P I1 2

2I

2 21 A f x 1 A f x

1 2 2 2x P[n]

22 2 A f x 1I24 2x P[n]

1I2

Pr I T(x, y ,y ) E

E 1 1 A f

1 f

variation

1 2

11 2 I2

w P Iz ,z P I

Pr I T(w, z ,z ) 1 f

variation

1 2

11 2 I2

w P Iz ,z P I

Pr I T(w, z ,z ) 1 f

variation

©©S.SafraS.Safra

Junta TestJunta Test

The junta-size test The junta-size test JTJT on a on a Boolean function Boolean function ff is is Randomly partition Randomly partition [n][n] to to II11, .., I, .., Irr

Run the independence-test Run the independence-test tt times on each times on each IIhh

Accept if Accept if ≤j ≤j of the of the IIhh fail their fail their independence-testsindependence-tests

For For r>>jr>>j22 and and t>>jt>>j22//

©©S.SafraS.Safra

CompletenessCompleteness

LemmaLemma: for a : for a jj-junta -junta ff

ProofProof: : only those sets which only those sets which contain an index of the Junta contain an index of the Junta would fail the independence-testwould fail the independence-test

1 2

1 2x P Iy ,y P I

Pr J T(x, y ,y ) 1

1 2

1 2x P Iy ,y P I

Pr J T(x, y ,y ) 1

©©S.SafraS.Safra

SoundnessSoundness

LemmaLemma::

ProofProof: Assume the premise. Fix : Assume the premise. Fix <<1/t<<1/t and and letlet

iJ i | f influence iJ i | f influence

1 2

1 2x P Iy ,y P I

1Pr J T(x, y ,y ) 2

f ( , jis an j) unta

1 2

1 2x P Iy ,y P I

1Pr J T(x, y ,y ) 2

f ( , jis an j) unta

©©S.SafraS.Safra

|J| ≤ j|J| ≤ j

PropProp: : r >> jr >> j implies implies |J| ≤ j|J| ≤ j

ProofProof: otherwise,: otherwise,

JJ spreads among spreads among IIhh w.h.p. w.h.p.

and for any and for any IIhh s.t. s.t. IIhhJ ≠ J ≠ it must be that it must be that VariationVariationIIhh(f) > (f) >

©©S.SafraS.Safra

High Frequencies Contribute High Frequencies Contribute LittleLittle

PropProp: : k >> r log r k >> r log r implies implies

ProofProof: a character : a character SS of size larger than of size larger than kk spreads w.h.p. over all parts spreads w.h.p. over all parts IIhh, hence , hence contributes to the influence of all parts.contributes to the influence of all parts.If such characters were heavy (If such characters were heavy (>>/4/4), ), then surely there would be more than then surely there would be more than j j parts parts IIhh that fail the that fail the tt independence-tests independence-tests

22k

2S k

ff S 4

22k

2S k

ff S 4

©©S.SafraS.Safra

Almost all Weight is on Almost all Weight is on JJ

LemmaLemma::

ProofProof: assume by way of contradiction : assume by way of contradiction otherwiseotherwise

sincesince

for a random partition w.h.p. (Chernoff bound)for a random partition w.h.p. (Chernoff bound)for every for every hh

however, since for any however, since for any II

the influence of every the influence of every IIhh would be would be ≥ ≥ /100rk/100rk

kJ

f 4 Variation k

Jf 4

Variation

k ki J

i J

ff

Variation Variation k ki J

i J

ff

Variation Variation

k ki I

i I

f k f

Variation Variation k ki I

i I

f k f

Variation Variation

h

ki

i I

f 100r

Variation h

ki

i I

f 100r

Variation

©©S.SafraS.Safra

Find the Close Find the Close JuntaJunta

Now, sinceNow, since

consider the (non Boolean)consider the (non Boolean)

which, if rounded outside which, if rounded outside JJ

is Boolean and not more than is Boolean and not more than far from far from ff

2k kJ J 2

ff f 2 Variation Variation 2k k

J J 2ff f 2

Variation Variation

SS J

g f S

SS J

g f S

Jf ' x sign A f x J Jf ' x sign A f x J

©©S.SafraS.Safra

Consider the q-biased product distribution q:

DefDef: : The probability of a subset The probability of a subset FF

and for a family of subsets and for a family of subsets

Consider the q-biased product distribution q:

DefDef: : The probability of a subset The probability of a subset FF

and for a family of subsets and for a family of subsets

Product, Biased DistributionProduct, Biased Distribution

F n Fnq F q (1 q) F n Fnq F q (1 q)

nF q

n nq q

F

Pr F F

nF q

n nq q

F

Pr F F

©©S.SafraS.Safra

Beckner/Nelson/Bonami Beckner/Nelson/Bonami InequalityInequality

DefDef: let : let TT be the following operator on any be the following operator on any ff, ,

PropProp::

ProofProof::

1 / 2z

f x f x zE

T

1 / 2z

f x f x zE

T

S

SS n

ff S

T

SS

S n

ff S

T

S SS n z

f x f S x zE

T

S SS n z

f x f S x zE

T

©©S.SafraS.Safra

Beckner/Nelson/Bonami Beckner/Nelson/Bonami InequalityInequality

DefDef: let : let TT be the following operator on any be the following operator on any ff, ,

ThmThm: for any : for any p≥r p≥r andand ≤((r-1)/(p-1))≤((r-1)/(p-1))½½

1 / 2z

f x f x zE

T

1 / 2z

f x f x zE

T

rpff T

rpff T

©©S.SafraS.Safra

Beckner/Nelson/Bonami Beckner/Nelson/Bonami CorollaryCorollary

Corollary 1Corollary 1: for any real : for any real ff and and 2≥r≥1 2≥r≥1

Corollary 2Corollary 2: for real : for real ff and and r>2 r>2

k

2r2

r 1 fkf k

2r2

r 1 fkf

k

22r

r 1 fkf k

22r

r 1 fkf

©©S.SafraS.Safra

Average SensitivityAverage Sensitivity

The sum of variables’ influence is referred The sum of variables’ influence is referred to as the average sensitivityto as the average sensitivity

Which can be expressed by the Fourier Which can be expressed by the Fourier coefficients ascoefficients as

ii [n]

ff

as influence ii [n]

ff

as influence

2

S

ff (S) S as 2

S

ff (S) S as

©©S.SafraS.Safra

Freidgut TheoremFreidgut Theorem

ThmThm: any Boolean : any Boolean ff is an is an [[, j]-, j]-junta for junta for

ProofProof::1.1. Specify the junta Specify the junta JJ

2.2. Show the complement of Show the complement of JJ has little influence has little influence

f / O asj = 2 f / O asj = 2

©©S.SafraS.Safra

Specify the JuntaSpecify the Junta

Set Set k=k=(as(f)/(as(f)/)), and , and =2=2--(k)(k)

Let Let

We’ll prove:We’ll prove:

and letand let

hence, hence, JJ is a is a [[,j]-,j]-junta, and junta, and |J|=2|J|=2O(k)O(k)

iJ i | f influence iJ i | f influence

2

J 2A f 1 2

2

J 2A f 1 2

Jf ' (x) sign A f x J Jf ' (x) sign A f x J

©©S.SafraS.Safra

High Frequencies Contribute High Frequencies Contribute LittleLittle

PropProp: :

ProofProof: a character : a character SS of size larger than of size larger than kk contributes at least contributes at least kk times the times the square of its coefficient to the square of its coefficient to the average sensitivity.average sensitivity.If such characters were heavy If such characters were heavy ((>>/4/4), ), as(f)as(f) would have been large would have been large

22k

2S k

ff S 4

22k

2S k

ff S 4

©©S.SafraS.Safra

AltogetherAltogether

LemmaLemma: :

ProofProof:: Jf 2

influence Jf 2

influence

2k kJ J2

ff f 2 influence + influence 2k k

J J2ff f 2

influence + influence

©©S.SafraS.Safra

AltogetherAltogether

k kJ

i J

2 2

O(k)S S

i S,S k i Si J i J r2

4/ r

O(k)S

i Si J 2

22/ rO(k) O(k) r

i J

ff

f(S) 2 f(S)

2 f(S)

as f2 f 2

i

i

influence influence

influence

k kJ

i J

2 2

O(k)S S

i S,S k i Si J i J r2

4/ r

O(k)S

i Si J 2

22/ rO(k) O(k) r

i J

ff

f(S) 2 f(S)

2 f(S)

as f2 f 2

i

i

influence influence

influence

©©S.SafraS.Safra

BiasedBiased qq--InfluenceInfluence

The The qq-influence-influence of an index of an index i i [n][n] on a on a boolean function boolean function f:P[n] f:P[n] {1,-1}{1,-1} is is

nqx

(f ) Pr f x f x i

qiInfluence

nqx

(f ) Pr f x f x i

qiInfluence

q

2

i2

f 1 A f qiinfluence

q

2

i2

f 1 A f qiinfluence

n

qi 1

ff q

ias influence n

qi 1

ff q

ias influence

©©S.SafraS.Safra

ThmThm [Margulis-Russo]: [Margulis-Russo]:

For monotoneFor monotone ff

HenceHenceLemmaLemma::For monotoneFor monotone ff > 0 > 0, , q q[p, p+[p, p+]] s.t. s.t. asasqq(f) (f) 1/ 1/

ProofProof:: Otherwise Otherwise p+p+(f) > 1(f) > 1

qq

d (f )as (f )

dq

q

q

d (f )as (f )

dq

©©S.SafraS.Safra

ProofProof [Margulis-Russo]: [Margulis-Russo]:

i

n nq q q

i qi 1 i 1i

d ( ) ( )as ( )

dq q

influencei

n nq q q

i qi 1 i 1i

d ( ) ( )as ( )

dq q

influence

©©S.SafraS.Safra

InfluentialInfluential People and Issues People and Issues

The theory of the The theory of the influenceinfluence of variables on of variables on Boolean functionsBoolean functions [BL, KKL][BL, KKL] and related and related issues, has been introduced to tackle issues, has been introduced to tackle social choicesocial choice problems, furthermore has problems, furthermore has motivated a magnificent sequence of motivated a magnificent sequence of works, related to Economics [K], works, related to Economics [K], percolation [BKS], Hardness of percolation [BKS], Hardness of approximation [DS]approximation [DS]Revolving around the Revolving around the Fourier/Walsh Fourier/Walsh analysis of Boolean functionsanalysis of Boolean functions… …

And the real important question:And the real important question:

©©S.SafraS.Safra

Where to go for Dinner?Where to go for Dinner?

Who has suggestions:Who has suggestions:

Each cast their vote in an Each cast their vote in an (electronic) envelope, (electronic) envelope, and have the system and have the system decided, not necessarily decided, not necessarily according to majority…according to majority…

It turns out someone –in It turns out someone –in the Florida wing- has the the Florida wing- has the power to flip some votespower to flip some votes

PowerPower

influenceinfluence

©©S.SafraS.Safra

Voting SystemsVoting Systems nn agents, each voting either “for” ( agents, each voting either “for” (TT) )

or “against” (or “against” (FF) – a Boolean function ) – a Boolean function over over nn variables variables ff is the outcome is the outcome

The values of the agents (variables) The values of the agents (variables) may each, independently, flip with may each, independently, flip with probability probability

It turns outIt turns out: one cannot design an : one cannot design an ff that would be robust to such noise -that would be robust to such noise -that is, would, on average, change that is, would, on average, change value w.p. value w.p. < < O(1)O(1)- unless taking into - unless taking into account only very few of the votesaccount only very few of the votes

©©S.SafraS.Safra

[n][n]x

IIz

[n][n]

Noise-SensitivityNoise-Sensitivity

How often does the value of How often does the value of ff changes changes when the input is perturbed?when the input is perturbed?

x

IIz

©©S.SafraS.Safra

DefDef((,p,x,p,x[n][n] ): Let ): Let 0<0<<1<1, and , and xxP([n])P([n]). .

Then Then y~y~,p,x,p,x, if , if y = (x\I)y = (x\I) z z where where I~I~

[n][n] is a is a noise subsetnoise subset, and, and z~ z~ pp

II is a is a replacementreplacement..

DefDef((-noise-sensitivity-noise-sensitivity): let ): let 0<0<<1<1, then, then

[ When [ When p=½p=½ equivalent to flipping each equivalent to flipping each coordinate in coordinate in xx independently w.p. independently w.p. /2/2.].]

[n] [n]p ,p,xx~ ,y~

ns f = Pr f x f y

[n] [n]p ,p,xx~ ,y~

ns f = Pr f x f y

[n][n]xIIz

Noise-SensitivityNoise-Sensitivity

©©S.SafraS.Safra

Noise-Sensitivity – Cont.Noise-Sensitivity – Cont.

AdvantageAdvantage: very efficiently testable (using : very efficiently testable (using only two queries) by a only two queries) by a perturbation-testperturbation-test..

DefDef ((perturbation-testperturbation-test): choose ): choose x~x~pp, and , and y~y~,p,x,p,x, check whether , check whether f(x)=f(y)f(x)=f(y) The success is proportional to the noise-The success is proportional to the noise-sensitivity of sensitivity of ff..

PropProp: the : the -noise-sensitivity is given by -noise-sensitivity is given by

2S

S

2 ns f =1 1 f S 2S

S

2 ns f =1 1 f S

©©S.SafraS.Safra

Relation between Relation between ParametersParameters

PropProp: small : small nsns small small high-freq weighthigh-freq weight

ProofProof::therefore: therefore: if if nsns is small, then is small, then Hence the Hence the high frequencieshigh frequencies must must have small weights (ashave small weights (as ). ).

PropProp: small : small asas small small high-freq weighthigh-freq weight

ProofProof:: 2

S

ff (S) S as 2

S

ff (S) S as

2S

S

2 ns f =1 1 f S 2S

S

2 ns f =1 1 f S

2S

S

1 f S ~1 2S

S

1 f S ~1

2

S

f S 1 2

S

f S 1

©©S.SafraS.Safra

Main ResultMain Result

TheoremTheorem: :

constant constant >0>0 s.t. any Boolean function s.t. any Boolean function

f:P([n])f:P([n]){-1,1}{-1,1} satisfying satisfying

is an is an [[,j]-junta,j]-junta for for j=O(j=O(-2-2kk332k2k).).

CorollaryCorollary: :

fix a fix a pp-biased distribution -biased distribution pp over over P([n])P([n])

Let Let >0>0 be any parameter. be any parameter.

Set Set k=logk=log1-1-(1/2)(1/2)

Then Then constant constant >0>0 s.t. any Boolean function s.t. any Boolean function

f:P([n])f:P([n]){-1,1}{-1,1} satisfying satisfying

is an is an [[,j]-junta,j]-junta for for j=O(j=O(-2-2kk332k2k))

2k22

f Ok

2k22

f Ok

2

ns f O k 2

ns f O k

©©S.SafraS.Safra

First Attempt: First Attempt: Following Freidgut’s ProofFollowing Freidgut’s Proof

ThmThm: any Boolean function : any Boolean function ff is an is an [[,j]-,j]-junta junta for for

ProofProof::1.1. Specify the juntaSpecify the junta

where, let where, let k=O(as(f)/k=O(as(f)/)) and fix and fix =2=2-O(k)-O(k)

2.2. Show the complement of Show the complement of JJ has small has small variationvariation

f / O asj = 2 f / O asj = 2

iJ i | f variation iJ i | f variation

P([n])

J

©©S.SafraS.Safra

If If kk were 1 were 1

Easy caseEasy case (!?!): If we’d have a bound on the (!?!): If we’d have a bound on the non-linear weight, we should be done.non-linear weight, we should be done.

The linear part is a set of independent The linear part is a set of independent characters (the singletons)characters (the singletons)

In order for those to hit close to 1 or -1 most In order for those to hit close to 1 or -1 most of the time, they must avoid the Law of of the time, they must avoid the Law of Large Numbers, namely be almost entirely Large Numbers, namely be almost entirely placed on one singleton [by Chernoff like placed on one singleton [by Chernoff like bound]bound]

Thm[FKN, ext.]: Assume Thm[FKN, ext.]: Assume ff is close to is close to linear, linear, then then ff is is close to close to shallowshallow ( ( a constant a constant function or a dictatorship) function or a dictatorship)

©©S.SafraS.Safra

How to Deal with Dependency How to Deal with Dependency between Charactersbetween Characters

RecallRecall

(theorem’s premise)(theorem’s premise)

IdeaIdea: Let: Let Partition Partition [n]\J[n]\J into into II11,…,I,…,Irr, for , for r >> kr >> k w.h.p w.h.p ffII[x][x] is close to is close to linearlinear (low freq (low freq

characters intersect characters intersect II expectedly by expectedly by 11 element, while high-frequency weight is low).element, while high-frequency weight is low).

2k kJ J2

ff f variation + variation 2k kJ J2

ff f variation + variation

2k22

1f Ok

2k22

1f Ok

J i | f kivariation J i | f kivariation

P([n])

J

I1

I2IrI

©©S.SafraS.Safra

Shallow FunctionShallow Function

DefDef: a function : a function ff is is linearlinear, if only singletons , if only singletons have non-zero weight have non-zero weight

DefDef: a function : a function ff is is shallowshallow, if , if ff is either a is either a constant or a dictatorship.constant or a dictatorship.

ClaimClaim: Boolean linear functions are : Boolean linear functions are shallow.shallow.

0 1 2 3 k n

weight

Charactersize

©©S.SafraS.Safra

Boolean Linear Boolean Linear Shallow Shallow

ClaimClaim: Boolean linear functions are : Boolean linear functions are shallow.shallow.

ProofProof: let : let ff be Boolean linear be Boolean linear function, we next show:function, we next show:

1.1. {i{ioo}} s.t. s.t. ((i.e.i.e. ))

2.2. And conclude, that eitherAnd conclude, that either or or i.e.i.e. ff is shallow is shallow

0S , i ,f S 0 0S , i ,f S 0 00 iff fi 00 iff fi

ff ff 00 iffi 00 iffi

©©S.SafraS.Safra

Claim 1Claim 1 Claim 1Claim 1: let : let ff be Boolean linear be Boolean linear

function, then function, then {i{ioo}} s.t. s.t. ProofProof: w.l.o.g assume: w.l.o.g assume

for any for any zz{3,…,n}{3,…,n} consider consider xx0000=z, x=z, x1010=z=z{1}, x{1}, x0101=z=z{2}, x{2}, x1111=z=z{1,2}{1,2}

thenthen .. Next value must be far from Next value must be far from {-1,1}{-1,1} A contradiction! (boolean function) A contradiction! (boolean function) ThereforeTherefore

00 iff fi 00 iff fi

f 1 f 2 0 f 1 f 2 0

ab a'b'a,b a' ,b' : f x f x min f 1 , f 2 ab a'b'a,b a' ,b' : f x f x min f 1 , f 2

ab a'b'

ab a'b'1 1

ab a'b'2 2

f x f x

f 1 x x

f 2 x x

ab a'b'

ab a'b'1 1

ab a'b'2 2

f x f x

f 1 x x

f 2 x x

f 2 0 f 2 01

-1

?

©©S.SafraS.Safra

Claim 2Claim 2

Claim 2Claim 2: let : let ff be Boolean function, s.t. be Boolean function, s.t.

Then eitherThen either or or ProofProof: consider : consider f(f()) and and f(if(i00))::

ThenThen but but ff is Boolean, hence is Boolean, hence thereforetherefore

00 iff fi 00 iff fi

ff ff 00 iffi 00 iffi

0

0 0

ff fi

fi ffi

0

0 0

ff fi

fi ffi

0 0fi f 2 fi 0 0fi f 2 fi

0fi 0,1 0fi 0,1 0fi f 0,2 0fi f 0,2

1

-1

0 f f

0fi 0fi

0fi 0fi

©©S.SafraS.Safra

Proving FKN: Proving FKN: almost-linear almost-linear close to close to

shallowshallow TheoremTheorem: Let : Let f:P([n])f:P([n]) be be linearlinear, ,

LetLet let let ii00 be the index s.t. is maximal be the index s.t. is maximal

then then

NoteNote: : ff is is linearlinear, hence, hencew.l.o.g., assume w.l.o.g., assume ii00=1=1, then all we need to , then all we need to

show is:show is:

We show that in the following claim and We show that in the following claim and lemma.lemma.

0fi 0fi

2

2f 1

2

2f 1

0

2

0 i2

ff fi 1 o 1

0

2

0 i2

ff fi 1 o 1

n

ii 1

ff fi

n

ii 1

ff fi

n 2

i 2

fi 1 o 1

n 2

i 2

fi 1 o 1

©©S.SafraS.Safra

CorollaryCorollary

CorollaryCorollary: Let : Let ff be linear, and be linear, andthen then a a shallow booleanshallow boolean function function gg s.t.s.t.

ProofProof: let: let , let , let gg be the be the boolean function closest to boolean function closest to ll. . Then,Then,this is true, as this is true, as is small (by theorem),is small (by theorem), and additionallyand additionally is small, sinceis small, since

2f g 3 o 1 2f g 3 o 1

0ffi 0ffi

2

2f g 9 o 1 2

2f g 9 o 1

2

2f 1

2

2f 1

2l g

2l g

2fl

2fl

2

2f 1

2

2f 1

©©S.SafraS.Safra

Claim 1Claim 1

Claim 1Claim 1: Let : Let f f be linear. be linear. w.l.o.g., assumew.l.o.g., assumethen then global constant global constant c=min{p,1-p}c=min{p,1-p} s.t. s.t.

i 2,...,n: fi c i 2,...,n: fi c

f 1 f 2 ... f n f 1 f 2 ... f n

{} {1} {2} {i} {n} {1,2} {1,3} {n-1,n} S {1,..,n}

weight

Characters

Each of weight no more than Each of weight no more than cc

©©S.SafraS.Safra

Proof of Claim1Proof of Claim1

ProofProof: assume: assume for any for any zz{3,…,n}{3,…,n}, consider , consider

xx0000=z=z, , xx1010=z=z{1}{1}, , xx0101=z=z{2}{2}, , xx1111=z=z{1,2}{1,2}

thenthen Next value must be far from Next value must be far from {-1,1} {-1,1} !! A contradiction! (toA contradiction! (to ))

2

2f 1

2

2f 1

f 2 c f 2 c

ab a'b'a,b a' ,b' : f x f x min f 1 , f 2 c ab a'b'a,b a' ,b' : f x f x min f 1 , f 2 c

ab a'b'

ab a'b'1 1

ab a'b'2 2

f x f x

f 1 x x

f 2 x x

ab a'b'

ab a'b'1 1

ab a'b'2 2

f x f x

f 1 x x

f 2 x x

1

-1

?

©©S.SafraS.Safra

Where to go for Dinner?Where to go for Dinner?

Who has suggestions:Who has suggestions:

Each cast their vote in an Each cast their vote in an (electronic) envelope, (electronic) envelope, and have the system and have the system decided, not necessarily decided, not necessarily according to majority…according to majority…

It turns out someone –in It turns out someone –in the Florida wing- has the the Florida wing- has the power to flip some votespower to flip some votes

PowerPower

influenceinfluence

Of course they’ll have to discuss it

over dinner….