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 KindlerJoint work with Eldar Fischer & Guy KindlerJoint work with Eldar Fischer & Guy Kindler

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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.

n

f : P n T,F

f : 1,1 1,1

n

f : P n T,F

f : 1,1 1,1

x

f x f ' xPr x

f x f ' xPr

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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

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Functions as Inner-Product Functions as Inner-Product SpaceSpace

{-1

,1}

n{

-1,1

}n

ff

nn

2n2n{

-1,1

}n

{-1

,1}

n

ff

nn

2n2n

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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

nf : 1,1 nf : 1,1

S xS(x) 1 S xS(x) 1

Sx

f S E f x x S

xf S E f x x

SS

ff S SS

ff S

2n2n

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Simple ObservationsSimple Observations

ClaimClaim::

Thm [Parseval]: Thm [Parseval]:

Hence, for a boolean Hence, for a boolean ff

x

f E f(x)

xf E f(x)

S 2S 2

f (S) f S 2S 2

f (S) f

2 2

2S

f (S) f 1 2 2

2S

f (S) f 1

2n2n

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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

2n2n

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Fourier Representation of Fourier Representation of influenceinfluence

ProofProof: consider the : consider the II-average -average functionfunction

which in Fourier representation iswhich in Fourier representation is

andand

I

y P IA f (x) E f x I y

I

y P IA f (x) E f x I 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

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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

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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

2n2n

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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

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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

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

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

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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 all but Accept if all but ≤j ≤j of the of the IIhh fail fail their independence-teststheir independence-tests

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

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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

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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

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|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) >

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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

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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

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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 ,S k

g f S

SS J ,S k

g f S

y P J

f ' x g x J ymaj

y P J

f ' x g x J ymaj

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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.