Fourier Analysis,Fourier Analysis,Projections,Projections, Influences, Influences,

Juntas,Juntas,Etc…Etc…

Fourier Analysis,Fourier Analysis,Projections,Projections, Influences, Influences,

Juntas,Juntas,Etc…Etc…

©©S.SafraS.Safra

Boolean Functions and Boolean Functions and JuntasJuntas

A boolean functionA boolean function

DefDef: : ff is a is a jj-Junta-Junta if there exists if there exists JJ[n][n]wherewhere |J|≤ j |J|≤ j, and s.t. for every , and s.t. for every xx

f(x) = f(x f(x) = f(x J) J)

ff is is ((, j)-, j)-JuntaJunta if if jj-Junta -Junta f’f’ s.t. s.t.

f : P n 1,1 f : P n 1,1

x

f x f ' xPr x

f x f ' xPr

©©S.SafraS.Safra

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*

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

ff2n2n

**

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

Functions as anFunctions as anInner-Product Vector-SpaceInner-Product Vector-Space A functions A functions ff is a vector is a vector

Inner product (normalized)Inner product (normalized)

Norm (normalized) Norm (normalized)

n2f n2f

nx 2

f g f x g xE

nx 2

f g f x g xE

n

1/ pp

px 2

ff xE

n

1/ pp

px 2

ff 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 a boolean For a boolean ff

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

p

pf 1

p

pf 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 multiplicative functions, one Consider all multiplicative functions, one for each for each charactercharacter SS[n][n]

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 xS(x) 1 S xS(x) 1

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: (: (notnot normalized) normalized)

Thm [Parseval]: Thm [Parseval]:

Hence, for a boolean Hence, for a boolean ff

p p

p S n

ff S

p p

p 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

Simple ObservationsSimple Observations

ClaimClaim::

Hence, for any Hence, for any ff

x

f E f(x)

xf E f(x)

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*

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

Putting a Junta to the TestPutting a Junta to the TestPutting a Junta to the TestPutting a Junta to the Test

Joint work with Eldar Fischer & Guy KindlerJoint work with Eldar Fischer & Guy KindlerBuilding on [KKL,Freidgut,Bourgain]Building on [KKL,Freidgut,Bourgain]

Joint work with Eldar Fischer & Guy KindlerJoint work with Eldar Fischer & Guy KindlerBuilding on [KKL,Freidgut,Bourgain]Building on [KKL,Freidgut,Bourgain]

©©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

Variables` InfluenceVariables` Influence

The The influenceinfluence of an index of an index i i [n][n] on a boolean on a boolean function 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

i S

ff S

iInfluence 2

i S

ff S

iInfluence

©©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

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

kS

S k

ff S

kS

S k

ff S

kS

S k

ff S

©©S.SafraS.Safra

Subsets` InfluenceSubsets` Influence

DefDef: The : The influenceinfluence 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

Influence 2 2

I I2 S I

f 1 A ff S

Influence

2

k kI I

S IS k

ff f S

Influence Influence 2

k kI I

S IS k

ff f S

Influence Influence

©©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

influence

1 2

11 2 I2

w P Iz ,z P I

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

influence

©©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

influence

1 2

11 2 I2

w P Iz ,z P I

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

influence

1 2

11 2 I2

w P Iz ,z P I

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

influence

©©S.SafraS.Safra

Junta TestJunta Test

The junta-size test on a The junta-size test on a boolean function boolean function ff is isRandomly 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 I T(x, y ,y ) 1

1 2

1 2x P Iy ,y P I

Pr I 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 I T(x, y ,y

is an j

) 2

f ( , j) unta

1 2

1 2x P Iy ,y P I

1Pr I T(x, y ,y

is an j

) 2

f ( , 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 influenceinfluenceII(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: otherwise,: otherwise,sincesince

for a random partition w.h.p. (Chernoff for a random partition w.h.p. (Chernoff bound)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 influence k

Jf 4

influence

k ki J

i J

ff

influence influence k ki J

i J

ff

influence influence

k ki I

i I

f k f

influence influence k ki I

i I

f k f

influence influence

h

ki

i I

f 100r

influence h

ki

i I

f 100r

influence

©©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 influence influence 2k k

J J 2ff f 2

influence influence

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

Open ProblemsOpen Problems

Is there a characterization, via Is there a characterization, via Fourier transform, of all efficiently Fourier transform, of all efficiently testable properties?testable properties?

What about tests that probe What about tests that probe ff only at only at two or three points? With two or three points? With applications to hardness of applications to hardness of approximation.approximation.

©©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

CorollaryCorollary: for : for ff s.t. s.t. ff>k>k=0 =0 andand p≥r≥1 p≥r≥1

ProofProof::

k2

p r

p 1f

r 1

fk

2

p r

p 1f

r 1

f

SkSp

S np

pr

f S Tf f

f

SkSp

S np

pr

f S Tf f

f

SS rp

S

k

pn p

f f S T f

f

SS rp

S

k

pn p

f f S T f

f

SkSp rp

S n p

ff S T f

f

SkSp rp

S n p

ff S T f

f

©©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=O(as(f)/k=O(as(f)/)), and , and =2=2-O(k)-O(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 2f 1 2

avg2

J 2f 1 2

avg

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

©©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 contributes kk times the square of times the square of its coefficient to the average its coefficient to the average sensitivity.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

i J

2 2

O(k)S S

i S,S k i S,S ki J i J2 r

2 4/ r

O(k) O(k)S S

i S i Si J i

kJ

Jr 2

22/ rO(k) O(k) r

i J

f f

f (S) 2 f(S)

2 f(S) 2 f(S)

as f2 f 2

i

i

influence

inf

influenc

e

e

luenc

2 2

O(k)S S

i S,S k i S,S ki J i J2 r

2 4/ r

O(k) O(k)S S

i S i S

k kJ

i J

i J i Jr 2

22/ rO(k) O(k) r

i J

f (S) 2 f(S)

2 f(S) 2 f(S)

as f2 f

ff

2

i

i

influence i

influ

nflu

ence

ence

2

O(k)S

i S,S ki J r

2 4/ r

O(k) O(k)S S

i S i Si J i Jr

k kJ

i J

2

Si S,S ki J

2

22/ rO(k) O(k) r

i

2

J

ff

2 f(S)

2 f(S) 2 f(S)

as f2 f 2

f(S)

i

i

influenc

influenc

e inf e

e

luenc

2 4/ r

O(k) O(k)S S

i S i Si J

k kJ

i J

2 2

O(k)S S

i S,S

i Jr 2

22/ rO(k) O(k) r

k i S,S ki J i J2 r

i J

2 f

ff

f(S) 2 f(S)

(S) 2 f(S)

as f2 f 2

i

iinfluence influence

influence

k kJ

i J

2 2

O(k)S S

i S,S k i S,S ki J i J2 r

2

O(k)S

i Si

4/ r

O(k)S

i Si J 2

22/ rO(k) O(k) r

i J

J r

ff

f(S) 2 f(S)

2 f(S 2 f(S)

as f2 f 2

)

i

i

influence

in

influ

fluence

ence

k kJ

i J

2 2

O(k)S S

i S,S k i S,S ki J i J2 r

2 4/ r

O(k) O(k)S S

i S i Si J

22/ rO(k) O(k) r

i

J

J

ir 2

ff

f(S) 2 f(S)

2 f(S) 2 f

as f2

)

f 2

(S

i

iinfluence influence

influence

k kJ

i J

2 2

O(k)S S

i S,S k i S,S ki J i J2 r

2 4/ r

O(k) O(k)S S

i S i Si J i Jr 2

2/ rO(k)

i

(k)

J

2O r

ff

f(S) 2 f(S)

2 f(S) 2 f(S)

2 fas f

2

i

i

influence influence

influence

k kJ

i J

2 2

O(k)S S

i S,S k i S,S ki J i J2 r

2 4/ r

O(k) O(k)S S

i S i Si J i Jr 2

22/ rO(k) O(k) r

i J

ff

f(S) 2 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 q

iInfluence nqx

(f ) Pr f x f x i q

iInfluence

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

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

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

qq

d ( )as ( )

dq

q

q

d ( )as ( )

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

Erdös-Ko-RadoErdös-Ko-Rado

DefDef:: A family of subsets A family of subsets P[R] P[R] is is tt-intersecting if for every-intersecting if for everyFF11, F, F22 ,, |F |F11 F F22| | t t

ThmThm[Wilson,Frankl,Ahlswede-Khachatrian]:[Wilson,Frankl,Ahlswede-Khachatrian]:For a For a tt-intersecting -intersecting ,,

wherewhere

CorollaryCorollary: : pp(() > P) > P is not is not 22-intersecting-intersecting

p p i,ti

( ) max (A ) p p i,ti

( ) max (A )

i,tA F | F 1,...,2i t i t i,tA F | F 1,...,2i t i t

p p i,2i

( ) max (A ) p p i,2i

( ) max (A ) PP = =