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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
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 = =