Fourier Analysis,
Projections,
Influences,
Juntas,
Etc…
Boolean Functions and Juntas
A boolean function
f : P n  1,1
Def: f is a j-Junta if there exists J[n]
where |J|≤ j, and s.t. for every x
f(x) = f(x  J)

©S.Safra
f is (, j)-Junta if  j-Junta f’ s.t. Pr f  x   f' x   
x
Functions as an
Inner-Product Vector-Space
11*
1*
10*
*
01*
0*
00*
111*
110*
101*
100*
011*
010*
001*
000*
©S.Safra
f
2n
11*
11*
Functions as an
Inner-Product Vector-Space

A functions f is a vector
2n
f

Inner product (normalized)
fg

x2n
Norm (normalized)
fp
©S.Safra
E f  x   g  x 
p 1/p
 fx 
E

n 
x2
1*
1*
10*
10*
**
01*
01*
0*
0*
00*
00*
111*
111*
110*
110*
101*
101*
100*
100*
011*
011*
010*
010*
001*
001*
000*
000*
ff
11*
11*
1*
1*
10*
10*
Simple Observations
Claims:
f 1  E  f(x) 
x
For a boolean f
p
fp 1
©S.Safra
**
01*
01*
0*
0*
00*
00*
111*
111*
110*
110*
101*
101*
100*
100*
011*
011*
010*
010*
001*
001*
000*
000*
ff
11*
11*
1*
1*
10*
10*
Fourier-Walsh Transform

©S.Safra
00*
00*
Sx
Given any function f : P n 
let the Fourier-Walsh coefficients of f be
f  S   f  S

01*
01*
0*
0*
Consider all multiplicative functions, one for
each character S[n]
S (x)  1

**
thus f can described as
f   f SS
S
111*
111*
110*
110*
101*
101*
100*
100*
011*
011*
010*
010*
001*
001*
000*
000*
ff
11*
11*
1*
1*
10*
10*
Fourier Transform: Norm
Norm: (not normalized)
f
Thm [Parseval]:
p
p
 f S
01*
01*
0*
0*
p
Sn
f  f2
Hence, for a boolean f
2
2
 f (S) 
S
©S.Safra
**
2
f2 1
00*
00*
111*
111*
110*
110*
101*
101*
100*
100*
011*
011*
010*
010*
001*
001*
000*
000*
ff
11*
11*
1*
1*
10*
10*
Simple Observations
Claim:

Hence, for any f
00*
00*
011*
011*
010*
010*
001*
001*
000*
000*
x
V   f  E  f
2
xP n
xP n
2
f 2  f   
©S.Safra
01*
01*
0*
0*
101*
101*
100*
100*
f     E f(x) 

2
**
111*
111*
110*
110*
2
 x   E f  x   

Sn,S 
2
f S 
ff
Putting a Junta to the Test
Joint work with Eldar Fischer & Guy Kindler
Building on [KKL,Freidgut,Bourgain]
Junta Test
Def: A Junta test is as follows:
A distribution over l queries  : P n  0,1
For each l-tuple, a local-test that either accepts or
rejects:
T[x1, …, xl]: {1, -1}l{T,F}
l
s.t. for a j-junta f
Prx1 ,..,xl T x1 ,.., xl   f   1
whereas for any f which is not (, j)-Junta
Prx1 ,..,xl T  x1 ,.., xl  (f)   1 2
©S.Safra
Variables` Influence

The influence of an index i [n] on a boolean
function f:{1,-1}n {1,-1} is


Influencei (f)  Pr f  x   f x  i 

xPn  

Which can be expressed in terms of the
Fourier coefficients of f
Claim:
2
Influence i  f    f  S 
iS
©S.Safra
Fourier Representation of influence
Proof: consider the I-average function on P[I]
AI f (x)  E f  x  y 
yPI
which in Fourier representation is
and
AI f 

SI 
f(S) S
2
©S.Safra
2
influencei  f   1  Ai f   f (S)
2
iS
High vs Low Frequencies
Def: The section of a function f above k is
f 
k
 f S
S k
S
and the low-frequency portion is
f 
k
 f S
S k
©S.Safra
S
Subsets` Influence
Def: The influence of a subset I  [n] on a
boolean function f is
InfluenceI  f  1  AI f
2
2
2

 f S
SI 
and the low-frequency influence
Influence
©S.Safra
k
I
 f  InfluenceI  f   
k
SI 
S k
2
f S 
Independence-Test
The I-independence-test on a boolean function
w  I, z1 ,z2  I
f is, for
IT(w, z1 ,z2 ) :
?
f  w  z1   f  w  z2 
Lemma:
Pr IT(w, z1 ,z2 )  1  21 influenceI  f 
wP I 
 
z1 ,z2 PI
©S.Safra
Pr IT(w, z1 ,z2 )  1  21 influenceI  f 
wP I 
 
z1 ,z2 PI

Pr IT(x, y1 ,y
,y2 )  E 
xP[n]
P[n] 
xP I 
y1 ,y2 PI


1AI f x 
2
 
2

1 AI f x 
2
2


1
 E 

1

1

A
f



I
2



2
xP[n] 
 1  21 influence I  f 
2
22 AI f x  
4
©S.Safra

2


Junta Test
 The
junta-size test on a
boolean function f is
 Randomly
partition [n] to I1, .., Ir
 Run the independence-test t
times on each Ih
 Accept if ≤j of the Ih fail their
independence-tests
For r>>j2 and t>>j2/
©S.Safra
Completeness
Lemma: for a j-junta f
Pr IT(x, y1,y2 )  1
xP I 
 
y1 ,y2 PI
Proof: only those sets which contain
an index of the Junta would fail
the independence-test
©S.Safra
Soundness
Lemma:
Pr IT(x, y1 ,y2 )  1
2
xP I 
y1 ,y2 PI 

f is an
( , j)  junta
Proof: Assume the premise. Fix <<1/t and let
J  i| influencei  f    
©S.Safra
|J| ≤ j
Prop: r >> j implies |J| ≤ j
Proof: otherwise,
J spreads among Ih w.h.p.
and for any Ih s.t. IhJ ≠  it must be that
influenceI(f) > 
©S.Safra
High Frequencies Contribute Little
Prop: k >> r log r implies
f
k 2
2

2

f
S




S k
4
Proof: a character S of size larger than k
spreads w.h.p. over all parts Ih, hence
contributes to the influence of all parts.
If such characters were heavy (>/4), then
surely there would be more than j parts Ih
that fail the t independence-tests
©S.Safra
Almost all Weight is on J
Lemma: influencek  f  
J
4
Proof: otherwise,
k
k
since
influencei  f  influenceJ  f 

iJ
for a random partition w.h.p. (Chernoff bound)
k
for every h
influence  f   

iIh
i
100r
however, since for any I
 influence  f  k  influence  f
iI
k
i
k
I
the influence of every Ih would be ≥ /100rk
©S.Safra
Find the Close Junta
Now, since
influenceJ  f   influence
k
J
 f 
f
k 2
2

consider the (non boolean)
g
 f S
S J
S
which, if rounded outside J
f' x   sign AJ f x  J  
is boolean and not more than  far from f
©S.Safra
2
Open Problems


©S.Safra
Is there a characterization, via Fourier
transform, of all efficiently testable
properties?
What about tests that probe f only at
two or three points? With applications
to hardness of approximation.
Product, Biased Distribution
Consider the q-biased product
distribution q:
Def: The probability of a subset F
 F   q  (1  q)
F
n
q
n F
and for a family of subsets 
qn    Pr F     qn F 
n
Fq
©S.Safra
F
Beckner/Nelson/Bonami
Inequality
Def: let T be the following operator on any f,
T f  x 
Prop:
Proof:
©S.Safra
T f 
T f  x  
E

z 1   / 2
f  x z 
  f S 
S
Sn 
 f S 
Sn 
S
S
 x E S z 
z
Beckner/Nelson/Bonami
Inequality
Def: let T be the following operator on any f,
T f  x 
E

z 1   / 2
f  x z 
Thm: for any p≥r and ≤((r-1)/(p-1))½
T f  f r
p
©S.Safra
Beckner/Nelson/Bonami Corollary
Corollary: for f s.t. f>k=0 and p≥r≥1
k
f
p
p 1


 r 1 
2
fr
Proof:
 fp 
k
©S.Safra
   f S 
S
Sn 
S
p
 T  f  p  f r
Average Sensitivity
The sum of variables’ influence is referred to
as the average sensitivity
as  f  
 influence  f
i
i[n]
Which can be expressed by the Fourier
coefficients as
2
as  f    f (S) S
S
©S.Safra
Freidgut Theorem
Thm: any boolean f is an [, j]-junta for
O as f /  
j =2
Proof:
1.
2.
©S.Safra
Specify the junta J
Show the complement of J has little influence
Specify the Junta
Set k=O(as(f)/), and =2-O(k)

 
Let J  i| influencei f  
We’ll prove:
and let

2
avgJ f  1  
2
2
f'(x)  sign  avgJ f  x  J  
hence, J is a [,j]-junta, and |J|=2O(k)
©S.Safra
High Frequencies Contribute Little
Prop:
f
k 2
2

2

f
S




S k
4
Proof: a character S of size larger than
k contributes k times the square of its
coefficient to the average sensitivity.
If such characters were heavy (>/4),
as(f) would have been large
©S.Safra
Altogether
Lemma: influenceJ  f   
2
Proof:
influenceJ  f   f
©S.Safra
k 2
2
+ influence Jk  f   
2
Altogether


k
k
k
k
influence
f

influence
influ
ence
influence JJ    iinf
luence ii  f  
nflu
ii
J
J


ii
S,
S, S
S
k
k
J
J ii
2
O(k)
O(k)
2
O(k)
O(k)
©S.Safra

ii
J
J
2
2
f(S) SS
2
2
f(S
f(S)) 
 f(S)
ii
S
S
O(k)
 2O(k)

ii
S,
S, S
S
k
k
J
J ii
2
2
S
S
2
O(k)
O(k)
rr
   influence
in
influenc
influ
encee ii  f  
inffluence
luenc
2/
2/ rr
ii
J


ii
J
J
2
O(k)
O (k)
O(k)
2
2
f(S) SS
f(S) 
 f(S)
ii
S
S

rr
4
4 // rr
S
S

2
2
as  f 
2r
r




Biased q-Influence

The q-influence of an index i [n] on a boolean
function f:P[n] {1,-1} is
Influenceiq (f)  Prn f  x   f  x 
i 
xq
influenceiq  f
asq  f 
©S.Safra
1
n

q
Ai f
q
influence

i  f
i1
2
2
Thm [Margulis-Russo]:
For monotone 
asq ( ) 
d q ( )
dq
Hence
Lemma:
For monotone 
 > 0,  q[p, p+] s.t. asq()  1/
Proof: Otherwise p+() > 1
©S.Safra
Proof [Margulis-Russo]:
dq ( )
dq
©S.Safra
n

i1
qi ( )
qi
n
  influence  asq ( )
i1
q
i
Erdös-Ko-Rado


Def: A family of subsets
  P[R] is t-intersecting if for every
F1, F2  , |F1  F2|  t
p (P) = max p (Ai,2 )
i
Thm[Wilson,Frankl,Ahlswede-Khachatrian]:
For a t-intersecting ,
where

©S.Safra
p ( )  max p (Ai,t )
i


Ai,t  F | F  1,...,2i  t  i  t
Corollary: p() > P   is not 2-intersecting