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.