Post on 21-Dec-2015
Fourier Analysis,Fourier Analysis,Projections,Projections, Influence, Influence,
Junta,Junta,Etc…Etc…
Fourier Analysis,Fourier Analysis,Projections,Projections, Influence, Influence,
Junta,Junta,Etc…Etc…
©©S.SafraS.Safra
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
x P n
ff xE
q
x P n
ff xE
qx P n
ff x
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
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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
©©S.SafraS.Safra
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
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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
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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
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….