Noise-Insensitive Noise-Insensitive Boolean-Functions Boolean-Functions are Juntas are Juntas

Noise-Insensitive Noise-Insensitive Boolean-Functions Boolean-Functions

are Juntas are Juntas

Guy Kindler & Muli Safra Guy Kindler & Muli Safra

Slides prepared with help of: Adi Akavia

Slides prepared with help of: Adi Akavia

Influential

Influential People People

 The theory of the The theory of the Influence Influence of Variables of Variables on Boolean Functions

on Boolean Functions [KKL,BL,R,M] [KKL,BL,R,M] and and related issues, has been introduced to related issues, has been introduced to

tackle

tackle social choice social choice problems. This area problems. This area has motivated a magnificent sequence of has motivated a magnificent sequence of

works, related to E

works, related to E conomics conomics [K], [K], percolation

percolation [BKS], [BKS], Hardness of Hardness of Approximation

Approximation [DS] [DS]

Revolving around the

Revolving around the Fourier/Walsh Fourier/Walsh analysis of Boolean functions

analysis of Boolean functions … …

 And the real important question: And the real important question:

Where to go for Dinner?

Where to go for Dinner?

The The alternatives alternatives

Diners would cast their vote in Diners would cast their vote in

an (electronic) envelope.

an (electronic) envelope.

The system would decide – The system would decide –

not necessarily by majority…

not necessarily by majority…

It turns out someone –in the It turns out someone –in the

Florida wing- has the ability Florida wing- has the ability

to flip some votes to flip some votes

Power Power

influence

influence

Voting Systems Voting Systems

 n n agents, each voting either “for” ( agents, each voting either “for” ( T T ) or ) or

“against” (

“against” ( F F ) – a Boolean function over ) – a Boolean function over n n variables

variables f f is the outcome is the outcome

 The values of the agents (variables) may The values of the agents (variables) may each, independently, flip with probability each, independently, flip with probability

 

 Bottom Line Bottom Line : one cannot design an : one cannot design an f f that that would be robust to such noise --that is, would be robust to such noise --that is,

would, on average, change value w.p.

would, on average, change value w.p.

<

<   ^{O(1) O(1)} -- unless taking into account only -- unless taking into account only very few of the votes

very few of the votes

Dictatorship Dictatorship

Def Def : a Boolean function : a Boolean function P([n]) P([n])   {-1,1} {-1,1} is a is a monotone

monotone e e - - dictatorships dictatorships --denoted --denoted f f e _e -- -- if: if:

e  

T e x

f x F e x

 

   

e  

T e x

f x F e x

 

   

Juntas Juntas

Def Def : a Boolean function : a Boolean function f:P([n]) f:P([n])   {-1,1} {-1,1} is a is a j j - - Junta Junta if if   J J   [n] [n] where where |J|≤ j |J|≤ j , ,

s.t. for every

s.t. for every x x   P([n]) P([n]) , , f(x) = f(x f(x) = f(x   J) J) Def Def : : f f is an is an [ [   , j] , j] - - Junta Junta if if

  j- j- Junta Junta f’ f’ s.t. s.t.

Def Def : : f f is an is an [ [   , j, p] , j, p] - - Junta Junta if if

  j- j- Junta Junta f’ f’ s.t. s.t.

   

x~Un

f x f' x

Pr ^{          }

x~Un

f x f' x

Pr ^{      }

   

x~ p

f x f' x

Pr

_

^{          }

x~ p

f x f' x

Pr

_

^{      }

We would tend to omit p

p-biased, product distribution

Long-Code Long-Code

 In the long-code In the long-code L:[n] L:[n]   {0,1} {0,1}

²²ⁿⁿ

each element is each element is encoded by an

encoded by an 2 2

ⁿⁿ

-bits -bits

 This is the most extensive binary code, having This is the most extensive binary code, having one bit for every subset in

one bit for every subset in P([n]) P([n])

Long-Code Long-Code

 Encoding an element Encoding an element e e   [n] [n] : :

 E E

_ee

legally-encodes legally-encodes an element an element e e if if E E

_ee

= f = f

_ee

F F F F T T T T T T

Long-Code

Long-Code   Monotone-Dictatorship Monotone-Dictatorship

 The truth-table of a Boolean function The truth-table of a Boolean function over

over n n elements, can be considered as a elements, can be considered as a 2 2 ^{n n} bits long string (each corresponding bits long string (each corresponding

to one input setting – or a subset of

to one input setting – or a subset of [n] [n] ) )

 For a long-code, the legal code-words For a long-code, the legal code-words are all monotone dictatorships

are all monotone dictatorships

 How about the Hadamard code? How about the Hadamard code?

Long-code Tests Long-code Tests

 Def Def (a (a long-code test long-code test ): given a code- ): given a code- word

word w w , probe it in a constant number of , probe it in a constant number of entries, and

entries, and

 accept w.h.p if accept w.h.p if w w is a monotone dictatorship is a monotone dictatorship

 reject w.h.p if reject w.h.p if w w is not close to any is not close to any monotone dictatorship

monotone dictatorship

Efficient Long-code Tests Efficient Long-code Tests

For some applications, it suffices if the test may For some applications, it suffices if the test may

accept illegal code-words, nevertheless, ones accept illegal code-words, nevertheless, ones

which have short

which have short list-decoding list-decoding : :

Def Def (a (a long-code list-test long-code list-test ): given a code-word ): given a code-word w w , , probe it in 2 or 3 places, and

probe it in 2 or 3 places, and



accept w.h.p if accept w.h.p if w w is a monotone dictatorship, is a monotone dictatorship,



reject w.h.p if reject w.h.p if w w is not even is not even approximately approximately

determined by a short list of domain elements determined by a short list of domain elements

that is, if that is, if   a a Junta Junta J J   [n] [n] s.t. s.t. f f is close to is close to f’ f’ and and f’(x)=f’(x

f’(x)=f’(x  J) J) for all for all x x

Note Note : a long-code list-test, distinguishes between the case : a long-code list-test, distinguishes between the case w is a w is a

dictatorship , to the case w is far from a is far from a junta .

Background Background



Thm (Friedgut) Thm (Friedgut) : a Boolean function : a Boolean function f f with small with small average-sensitivity

average-sensitivity is an is an [ [  ,j]- ,j]- junta junta



Thm (Bourgain) Thm (Bourgain) : a Boolean function : a Boolean function f f with small with small high- high- frequency weight

frequency weight is an is an [ [  ,j]- ,j]- junta junta



Thm Thm : a Boolean function : a Boolean function f f with small with small high-frequency high-frequency weight

weight in a in a p p - - biased biased measure is an measure is an [ [   ,j]- ,j]- junta junta



Corollary Corollary : a Boolean function : a Boolean function f f with with small small noise- _noise- sensitivity

sensitivity is an is an [ [   ,j]- ,j]- junta junta



Parameters Parameters : : average-sensitivity average-sensitivity [M,R,BL,KKL,F] [M,R,BL,KKL,F]

high-frequency weight

high-frequency weight [KKL,B] [KKL,B]

noise-sensitivity

noise-sensitivity [BKS] [BKS]

[n]

x [n]

z II

[n]

[n]

Noise-Sensitivity Noise-Sensitivity

How often does the value of

How often does the value of f f changes changes when the input is perturbed?

when the input is perturbed?

x

z II

 Def Def ( (   _{  ,p,x ,p,x} ^{[n] [n]} ): Let ): Let 0< 0<   <1 <1 , and , and x x   P([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 subset noise subset , and , and

 z~ z~  

_pp^II

is a is a replacement replacement . .

Def Def ( (   -noise-sensitivity -noise-sensitivity ): let ): let 0< 0<   <1 <1 , then , then

[ When

[ When p=½ p=½ equivalent to flipping each equivalent to flipping each coordinate in

coordinate in x x w.p. w.p.   /2 /2 .] .]

   

[n] [n]

p ,p,x

x~ ,y~

ns f = Pr f x f y



     

_[n]



_[n]

        

p ,p,x

x~ ,y~

ns f = Pr f x f y



           

[n]

[n] x z II

Noise-Sensitivity

Noise-Sensitivity

Fourier/Walsh Transform Fourier/Walsh Transform

Write

Write f:{-1, 1} f:{-1, 1}

ⁿⁿ

  {-1, 1} {-1, 1} as a polynomial as a polynomial What would be the monomials?

What would be the monomials?



For every set For every set S S   [n] [n] we have a monomial which is the we have a monomial which is the product of all variables in

product of all variables in S S (the only relevant (the only relevant powers are either

powers are either 0 0 or or 1 1 ) )

It now makes sense to consider the degree of

It now makes sense to consider the degree of f f or to break it or to break it according to the various degrees of the monomials..

according to the various degrees of the monomials..

( )

[ ]



_S

S n

f f S c

Í

= å ^{( )}

[ ]



_S

S n

f f S c

Í

= å

High/Low Frequencies High/Low Frequencies

Def Def : the : the high-frequency high-frequency portion of portion of f f : :

Def Def : the : the low-frequency low-frequency portion of portion of f f : : Def Def : the : the high-frequency-weight high-frequency-weight is: is:

Def Def : the : the low-frequency-weight low-frequency-weight is: is:

 

k S

S k

f

^

f S 



  ^{  }

k S

S k

f

^

f S 



  ^

 

k S

S k

f

^

f S 



  ^{  }

k S

S k

f

^

f S 



  ^

2

 

k 2

2 S k

f

^

f S



  ^

²

^{ }

k 2

2 S k

f

^

f S



  ^

2

 

k 2

2 S k

f

^

f S



  ^

²

^{ }

k 2

2 S k

f

^

f S



  ^

Low High-Frequency Weight Low High-Frequency Weight

Prop Prop : the : the   -noise-sensitivity can be expressed in Fourier -noise-sensitivity can be expressed in Fourier transform terms as

transform terms as

Prop Prop : Low : Low ns ns   Low Low high-freq weight high-freq weight Proof

Proof : By the above proposition, low noise-sensitivity : By the above proposition, low noise-sensitivity implies

implies

nevertheless,

nevertheless, f f being being {-1, 1} {-1, 1} function, by Parseval function, by Parseval formula (that the

formula (that the norm 2 norm 2 of the function and its of the function and its Fourier transform are equal) implies

Fourier transform are equal) implies

( )

^S

^

²

( )

S

1 - l f S ~ 1

å ^{( )}

^S

^

²

^{( )}

S

1 - l f S ~ 1

å

2

 

S

f S 1 

 ^

²

^{ }

S

f S 1 

 ^

 

^{S 2}

 

S

2 ns f =1 

_

      ^ 1   ^

^S

f S ^

²

^{ }

S

2 ns f =1 

_

      1   f S ^

Average and Restriction Average and Restriction

Def Def : Let : Let I I   [n], [n], x x   P([n]\I) P([n]\I) , , the the restriction function restriction function is is

Def Def : the : the average function average function is is

Note Note : :

     

I

I y P I

A f : P I

A f x E f x y



  

     

  

  

   



     

I

I y 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 y P I I

A f x E f x y



 

      

     

I y P I I

A f x E f x y



 

    

I [n]

x y

I [n]

x y y

y y y

Fourier Expansion Fourier Expansion

 Prop Prop : :

 Prop Prop : : _{I S}

S I

A f f(S) 

 

      ^

I S

S I

A f f(S) 

 

      ^

    ^{ }

I T S

S I T I S

f x f T  x 

  

 

  

 

  ^

    ^{ }

I T S

S I T I S

f x f T  x 

  

 

  

 

  ^

Influence

Influence /Variation /Variation

Def Def : the : the variation variation of of I I on on f f : :

Prop Prop : the following are equivalent : the following are equivalent definitions to the

definitions to the variation variation of of I I on on f f : :

  ^{2 2  }

I I 2

S I

f f A f f S



 

     

  ^

variation _I   _I ² ₂ ^{2  }

S I

f f A f f S



 

     

  ^

variation

  _{   } ^{ }  

I f x P I E var f x y y P I I

 

   

      

variation _I   _{  y P I   I} ^{ }  

x P I

f E var f x y

 

   

      

variation

Influence

_i

(f) = variation

_i

(f) = variation

_{i}

(f)

Low-frequencies Variation and a.s.

Low-frequencies Variation and a.s.

Def Def : the : the low-frequency variation low-frequency variation is: is:

Def Def : : the the average-sensitivity average-sensitivity is is And in Fourier representation:

And in Fourier representation:

Def Def : the : the low-frequency average-sensitivity low-frequency average-sensitivity is: is:

 

i

 

i [n]

f f



 

as   variation

i

 

i [n]

f f



 

as variation

 

²

S

f   f (S) S ^ as  

²

S

f   f (S) S ^ as

    ^{2  }

k k

I I

S I S k

f f f S

 

  

   ^

variation ^{I k}   variation ^I   ^{k 2  }

S I S k

f f f S

 

  

   ^

variation variation

    ²

i [n] S k

f f f (S) S

 

 

    ^

k k

as   variation i   ²

i [n] S k

f f f (S) S

 

 

    ^

k k

as variation i

Biased Walsh Product

Biased Walsh Product [Talagrand] [Talagrand]

Def Def : In the : In the p p -biased product distribution -biased product distribution  

_pp

, the , the probability of a subset

probability of a subset x x is is

 The usual The usual Fourier basis Fourier basis   is not orthogonal with is not orthogonal with respect to the

respect to the biased inner-product, biased inner-product,

 Hence, we use the Hence, we use the Biased Walsh Product Biased Walsh Product : :

 

^{x n x}

pn

x p (1 p)



_pⁿ

  x p (1 p) 

^x

 

^{n x}

   

^

 

p 1 p i x

x 1 p i x

p



   

      



i

 

p 1 p i x

x 1 p i x

p



   

      



i

  x

_i

  x



^S

^{  x }  

ⁱ

^{  x}



^S

  

Main Result Main Result

Theorem Theorem : :

  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( 

^-2-2

k k

³³

 

^2k2k

) ) . . Corollary

Corollary : : fix a

fix a p p -biased distribution -biased distribution  

_pp

over over P([n]) P([n]) . . Let Let   >0 >0 be any parameter. be any parameter.

Set Set k=log k=log

1-_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( 

^-2-2

k k

³³

 

^2k2k

) ) . .

 

k 2 2 2

f

^{k 22}

 O k   

²

f

^

 O k 

   

²

ns f

_

   O k   

²

ns f

_

 O k 

The The KKL/Freidgut KKL/Freidgut Framework Framework

Thm Thm : any Boolean function : any Boolean function f f is an is an [ [   ,j]- ,j]- junta for junta for Proof

Proof : :

1.1.

Specify the junta Specify 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 J J has small variation has small variation

   ^{f / } 

j = 2 ^{O as} O as ^{   f /  }

j = 2



i

  

J   i| variation

i

  f   

J  i| variation f  

[n]

J

KKL/Freidgut KKL/Freidgut

Lemma Lemma : :

Proof Proof : :

Now, lets bound each argument:

Now, lets bound each argument:

Prop Prop [KKL] [KKL] : : Proof

Proof : characters of size : characters of size   k k contribute to the contribute to the average-sensitivity

average-sensitivity at least _{at least} (since

(since ) )

 

k 2 2

f

^k ²

 as f   k

2

f

^

 as f k

J

  f _  2 variation

_J

  ^{f  } ₂

variation

 

^k

 

^{k 2}

J

f 

J^

f  f

^ 2

variation

_J

  ^{f } variation

_J^k

  ^{f  f}

^{k 2}₂

variation variation

[n]

J

k 2

k f 

^{k 2}2

k f 

^ 2

 

²

^{ }

S

as f ^{ }   f S S ^

²

^{ } 

S

as f   f S S ^ 

Beckner/Nelson/Bonami

Beckner/Nelson/Bonami Inequality Inequality

Def Def : let : let T T   be the following operator on be the following operator on f f

Thm Thm : for any : for any p≥r p≥r and and   ≤((r-1)/(p-1)) ≤((r-1)/(p-1)) ^{½ ½}

Corollary

Corollary : for : for g g of degree of degree k k

   

1 ,p,x

y

f x E f y



  

_

 

      

T    

1 ,p,x

y

f x E f y



  

_

 

       T

p r

f f

    

T _r

f p f

     T

4 4k 4

4 2

g ⁴ ₄   ^{ 4k} g ₂ ⁴

g   ^ g

   

    ^{ }

k i J

2 2

S O(k) S

i S, S k i S, S k

i J 2 i J r

2 4/r

O(k) O(k)

S S

i S i S

i J r i J 2

2/r 2 O(k)

k J

O(k) r

f

f(S) 2 f(S)

2 f(S) 2 f(S)

2 f 2 as f

f

^



   

 

 

 



 

   

    

    



   

   



 

 

 

 

 

i

influenc

i

variation vari on

e

  ati  

    ^{ }

k i J

2 2

S O(k) S

i S, S k i S, S k

i J 2 i J r

2 4/r

O(k) O(k)

S S

i S i S

i J r i J 2

2/r 2 O(k)

k J

O(k) r

f

f(S) 2 f(S)

2 f(S) 2 f(S)

2 f 2 as f

f

^



   

 

 

 



 

   

    

    



   

   



 

 

 

 

 

i

influenc

i

variation vari on

e

ati

Freidgut

Freidgut ’s Proof ’s Proof

Prop Prop : : Proof Proof : :

 

k

J^

f _  4 variation

_J^k

  ^{f  } ₄

variation

we do not know

whether as(f) is small!



True only since this is a {-1,0,1} function.

So we cannot proceed this way with only

this way with only as as

^kk

!  !

If If k k were 1 were 1

Easy case

Easy case (!?!): If we’d have a bound on the non- (!?!): If we’d have a bound on the non- linear weight, we should be done.

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) Concentration of measure

Concentration of measure : In order for those to : In order for those to hit close to

hit close to 1 1 or or -1 -1 most of the time, they must most of the time, they must avoid the law of large numbers, namely be

avoid the law of large numbers, namely be almost entirely placed on one singleton [by almost entirely placed on one singleton [by

Chernoff like bound]

Chernoff like bound]

(!) (!) [FKN, ext.] [FKN, ext.] if if f f is close to is close to linear linear then then f f is is close to

close to shallow shallow ( (   a constant function or a a constant function or a dictatorship)

dictatorship)

Almost Linear

Almost Linear   Almost Shallow Almost Shallow

Thm( Thm( [FKN] [FKN] ) ) : :   global constant global constant M M , , s.t.

s.t.   Boolean function Boolean function f f , ,

  shallow shallow Boolean function Boolean function g g , s.t. , s.t.

 Hence, Hence, ||f ||f _{I I} [x] [x] ^{>1 >1} || || _{2 2} is small is small     f f _{I I} [x] [x] is is close to

close to shallow shallow ! !

2 1 2

2 2

f g  ² ₂  M f  ^{ 1 2}

f g   M f  ^ 2

How to Deal with Dependency How to Deal with Dependency

between Characters?

between Characters?

Recall Recall

(theorem’s premise) (theorem’s premise)

Idea Idea : Let : Let

 Partition Partition [n]\J [n]\J into into I I

1₁

,…,I ,…,I

r_r

, for , for r >> k r >> k

 w.h.p w.h.p f f

_II

[x] [x] is close to is close to linear linear (low freq (low freq characters intersect

characters intersect I I expectedly by expectedly by   1 1

element, while high-frequency weight is low).

element, while high-frequency weight is low).

 

^{k 2 k}

 

J

f  f

^ 2 J^

f

variation

_J

  ^{f  f}

^{k 2}₂

+variation

_J^k

  ^f

variation +variation

 

k 2

2

1

2

f

^{k 22}

 O k   ¹

²

f

^

 O k

   

J   i| variation

^_i ^k

  f   

J  i| variation

^_i ^k

f  

[n]

J I

₁

I

₂

I

_r

I

So what?

So what?

f f I _I [x] [x] is close to is close to linear linear

By By [FKN] [FKN] , , f f I _I [x] [x] is shallow for any is shallow for any x x Still,

Still, f f I _I [x] [x] could be a different could be a different dictatorship for different

dictatorship for different x x ’s, hence ’s, hence the variation of each

the variation of each i i   I I might be low!! might be low!!

P([n])

J I

₁

I

₂

I

_r

I

Dictatorship and its Singleton Dictatorship and its Singleton

 Prop Prop : for a dictatorship : for a dictatorship h h , ,   coordinate coordinate i i s.t. s.t. (where (where p p is the bias). is the bias).

 Corollary (from [FKN]) Corollary (from [FKN]) : :   global constant global constant M M , s.t. , s.t.   Boolean function Boolean function h h , either , either

or or

 ( ) { }

h i  ( ) { } > p h i > p

    

h i       p h i  p

  ^{h  M h}

^{1 2}

variation   ^{h  M h}

^{1 2}

variation

{1} {2} {i} {n} {1,2} {1,3} {n-1,n} S {1,..,n}

weight

Characters Total weight of no more than

Total weight of no more than 1-p 1-p

Main Lemma Main Lemma

 Lemma Lemma : :     >0 >0 , s.t. for any , s.t. for any   and any and any function

function g:P([m]) g:P([m])     , the following , the following holds:

holds:

 ^m

   ^{4k k 2 4 k 2 2} 

x~ Pr g x O g g



^{ m}

^      ^     ^{ 4k  k 2 4}  ^{ k 2 2} 

x~ Pr g x O g g

 ^    ^    ^{ }  ^

Low-freq high-freq

Probability Concentration Probability Concentration

 Simple Bound Simple Bound : :

 Proof Proof : :

 Low-freq Bound Low-freq Bound : Let : Let g:P([m]) g:P([m])     be of be of degree

degree k k and and   >0 >0 , then , then     >0 >0 s.t. s.t.

 Proof Proof : recall the corollary: : recall the corollary:

 _m

  ^t

t

x~ Pr g x g t

 ^t 

^{ }_m

^      ^   ^t _t

x~ Pr g x g

  ^    ^  

 

 _m

  ^{4 4k} ₂ ⁴

x~ Pr g x g



^{ }_m

^      ^     ^{ 4  4k} ₂ ⁴

x~ Pr g x g

 ^    ^     ^{ }

4 4k 4

4 2

g ⁴ ₄   ^{ 4k} g ₂ ⁴

g   ^ g

Lemma’s Proof Lemma’s Proof

 Now, let’s prove the lemma: Now, let’s prove the lemma:

 Bounding low and high freq separately: Bounding low and high freq separately:  

  , ,

simple bound

 

4 2 2

4 4k k k

2 2

g g

  ^{  }   ^{ }

   ^{ 4  4k} g ^{ k} ₂ ⁴       ^{ 2} g ^{ k 2} ₂

  

 

 

 

 

_{ }

   

m

m m

x~

k k

x~ x~

Pr g x

Pr g x Pr g x



 



  

 

   

 

       

   

 

 

 

 

_{ }

   

m

m m

x~

k k

x~ x~

Pr g x

Pr g x Pr g x



 



  

 

   

 

       

   

Low-freq bound

f f _{I I} [x] [x] Mostly Constant Mostly Constant

 Lemma Lemma : :     >0 >0 , s.t. for any , s.t. for any   and any and any function

function g:P([m]) g:P([m])    

 Def Def : Let : Let D D _{I I} be the set of be the set of x x   P( P( I I ) ) , s.t. , s.t.

f f I _I [x] [x] is a dictatorship is a dictatorship

 Next we show, that Next we show, that |D |D I _I | | must be small, must be small, hence for most

hence for most x x , , f f _{I I} [x] [x] is constant. is constant.

  ^{      }

 

I I

D

^I

  x P I : i I,s.t. f x i      ^

^I

^{     }  p 

D  x P I : i I,s.t. f x i     p

 _m

 

₁ ^{4k k} ₂⁴ ₂ ^{k 2}₂

x~

Pr g x M g M g



^{ }_m

^      ^  

₁



^{4k k} ₂⁴



₂ ^{k 2}₂

x~

Pr g x M g M g

 ^    ^   

^{ }



^

 Lemma Lemma : :

 Proof Proof : denote : denote , then , then

|D |D _{I I} | | must be small must be small

 

[n]

4k k 2

I 1 2 2

x~

Pr x D M M f

_[n]

 

I

 

1

 

^4k



2 ^{k 2}2

x~

Pr x D M M f



   

^



^

  ^      

i I

g x

i

   f x i ^

I

     

g x  f x i

  _{ } ^{ }

 

   

 

 

 

[n]

i i

I i

x~ i I x P I

4 2

4k k k

1 i I 2 2 i I 2

4 2

1 4k S 2 S

i I S [n], S k,S I i 2 i I S [n], S k,S I i 2

2 2 2

4k k

1 2 2

i I S [n], S k,S I i i I

Pr x D Pr g x p

M g M g

M f S M f S

M f S M f





  



 

  

 



         

 

     

 

    

 

 

 

       



 

   

  

 



  _{ } ^{ }

 

   

 

 

 

[n]

i i

I i

x~ i I x P I

4 2

4k k k

1 2 2 2

i I i I

4 2

1 4k S 2 S

i I S [n], S k,S I i 2 i I S [n], S k,S I i 2

2 2 2

4k k

1 2 2

i I S [n], S k,S I i i I

Pr x D Pr g x p

M g M g

M f S M f S

M f S M f





  



 

  

 



         

 

     

 

    

 

 

 

       



 

   

  

 



Prev lemma

 

I

   

T T [n],

T I S

f x S f T



 





^

 

_I

   

_T

T [n], T I S

f x S f T



 





^

Each S is counted only for one index iI. (Otherwise, if S was counted for both i

and j in I, then |SI|>1!)

Parseval

Simple Prop Simple Prop

 Prop Prop : let : let {a {a i _i } } i _i   I I be be sub-distribution sub-distribution , that , that is, is,   _{i i  I I} a a i _i   1 1 , , 0 0   a a i _i , then , then   _{i i  I I} a a i _i 2 2   max max i _i  I I {a {a i _i } } . .

 Proof Proof : :

ⁱ² _max ^{max2 max}

i I

a 1 a a a



  



ⁱ² _max ^{max2 max}

i I

a 1 a a a



  



1 2 3 max n

a

_i

no more than no more than 1 1

1

a

_i

1/a 1/a

_maxmax

1

|D |D _{I I} | | must be small - Cont must be small - Cont

 Therefore Therefore

(since

(since ), ),

 Hence Hence

 

 

 

 

   

2

2 2 k

i I i I

i I S [n], S k, S [n], S k,

S I i S I i

f S max f S max

^

f 

 

    

   

   

 

      

 

 

 

   

  ^{  }  ^{ variation}

ⁱ

 

 

 

   

2

2 2 k

i I i I

i I S [n], S k, S [n], S k,

S I i S I i

f S max f S max

^

f 

 

    

   

   

 

      

 

 

 

   

  ^  ^{ variation}

ⁱ

 

 

2

S [n], S k,S I i

f S 1

   

 ^{  } 

 

2

S [n], S k,S I i

f S 1

   

 ^ 

 

[n]

4k k 2

I 1 2 2

x~ Pr x D M M f



_[n]

  _I   ₁   ^{ 4k}  ₂ ^{ k 2} ₂

x~ Pr x D M M f

     ^  ^

Where to go for Dinner?

Where to go for Dinner?

The The alternatives alternatives

Diners would cast their vote in Diners would cast their vote in

an (electronic) envelope.

an (electronic) envelope.

The system would decide – The system would decide –

not necessarily by majority…

not necessarily by majority…

It turns out someone –in the It turns out someone –in the

Florida wing- has the ability Florida wing- has the ability

to flip some votes to flip some votes

Power Power

influence influence

Of course they’ll have to discuss it over

dinner….

Discussion Discussion

 Tests that look at only 2 or 3 places Tests that look at only 2 or 3 places cannot produce a large gap between cannot produce a large gap between

probability of acceptance of a probability of acceptance of a

dictatorship and that of a function not dictatorship and that of a function not

so close to a junta so close to a junta

 Nevertheless, if requiring the function Nevertheless, if requiring the function to have additional properties, such as to have additional properties, such as

local-maximality, one may be able to local-maximality, one may be able to

design a test with a large gap

design a test with a large gap

Shallow Function Shallow Function

 Def Def : a function : a function f f is is linear linear , if only singletons , if only singletons have non-zero weight

have non-zero weight

 Def Def : a function : a function f f is is shallow shallow , if , if f f is either a is either a constant or a dictatorship.

constant or a dictatorship.

 Claim Claim : Boolean linear functions are shallow. : Boolean linear functions are shallow.

weight

Character

size

Boolean Linear

Boolean Linear   Shallow Shallow

 Claim Claim : Boolean linear functions are : Boolean linear functions are shallow.

shallow.

 Proof Proof : let : let f f be Boolean linear function, be Boolean linear function, we next show:

we next show:

1. 1.   {i {i

o_o

} } s.t. s.t.

( ( i.e. i.e. ) ₎

2. 2. And conclude, that either And conclude, that either or or i.e. i.e. f _f is shallow is shallow

  

0

 ^{ }

S  , i ,f S 0

  S   , i ,f S  

0

  ^{ }  0

  f f f f               f i f i          

00

  _{   }

ii₀₀

 

f f     

f f    f f i f f i            

00

  _{   }

ii₀₀

Claim 1 Claim 1

 Claim 1 Claim 1 : let : let f f be boolean linear function, be boolean linear function, then

then   {i {i o _o } } s.t. s.t.

 Proof Proof : w.l.o.g assume : w.l.o.g assume

 for any for any z z   {3,…,n} {3,…,n} , consider , consider

x x

00₀₀

=z =z , , x x

10₁₀

=z =z   {1} {1} , , x x

01₀₁

=z =z   {2} {2} , , x x

11₁₁

=z =z   {1,2} {1,2}

 then then . .

 Next value must be far from Next value must be far from {-1,1} {-1,1} , ,

 A contradiction! (boolean function) A contradiction! (boolean function)

 Therefore Therefore

     

0

_{ }

i₀

f f       f i     

0

 _{ }

i₀

f f     f i  

      ^{ }

f 1       f 2    ^{ }  0 f 1   f 2   0

  ^a,b  a',b' : f x  ^{ }

^ab

f x 

^a'b'

 min f 1 , f 2        ^{ } 

   ^a,b   a',b' : f x  ^{ }

^ab

 f x 

^a'b'

  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   0 ₁

-1

?

Claim 2 Claim 2

 Claim 2 Claim 2 : let : let f f be boolean function, s.t. be boolean function, s.t.

Then either

Then either or or

 Proof Proof : consider : consider f( f(   ) ) and and f(i f(i 0 ₀ ) ) : :

 Then Then

 but but f f is boolean, hence is boolean, hence

 therefore therefore

     

0

_{ }

i₀

f f       f i     

0

 _{ }

i₀

f f     f i  

f f       

f f    f f i f f i            

00

  _{   }

ii₀₀

       

    ^{ }   ^{ }

0

0 0

f f f i

f i f f i

 



 

 

 

 

       

    ^{ }   ^{ }

0

0 0

f f f i

f i f f i

 



 

 

 

 

   

0

^{ }   ^{ }

0

f i    

0

 f ^{ }    2 f i   ^{ }

0

f i  f    2 f i

   

0

^{ }

f i     

0

 ^{ } 0,1 f i   0,1

   

0

^{   }

f i    

0

 f ^{ }   ^{ } 0,2 f i  f   0,2

1

-1 0  

f   

f 

f if i

   

0₀

 

₀

f i

 

0

f i

Proving FKN:

Proving FKN:

almost-linear

almost-linear   close to shallow close to shallow

 Theorem Theorem : Let : Let f:P([n]) f:P([n])     be be linear linear , ,



Let Let



let let i i

0₀

be the index s.t. is maximal be the index s.t. is maximal then

then

 Note Note : : f f is is linear linear , hence , hence w.l.o.g., assume

w.l.o.g., assume i i

₀₀

=1 =1 , then all we need to show , then all we need to show is: is:

We show that in the following claim and lemma.

We show that in the following claim and lemma.

   

0

f i     

0

f i 

2

f 1

2

  f 1 

²₂

  

     

0

_{ }

i₀ ²

 ^{ } 

f  ^   f      f i     

0

 _{ }

i₀

^  

²2

  1 o 1  ^{ }   f  ^   f    f i   ^  

2

 1 o 1  

 

ⁿ

   

i i 1

f f  f i 



 ^{  }  

ⁿ

^{    }

ⁱ

i 1

f f  f i 



 ^   ^

     ^{ } 

n 2

i 2

f i 1 o 1 



 



ⁿ

^

²

^{       }

i 2

f i 1 o 1 



 

 ^

Corollary Corollary

 Corollary Corollary : Let : Let f f be linear, and be linear, and then

then   a a shallow boolean shallow boolean function function g g s.t. s.t.

 Proof Proof : let : let , let , let g g be the be the boolean function closest to

boolean function closest to l l . . Then,

Then,

this is true, as this is true, as

 is small (by theorem), is small (by theorem),

 and additionally and additionally is small, since is small, since

   

f g 

₂2

  3 o 1      f g   3 o 1  

     

0

f    f i 

  f      f i     

0

 

   

2

f g 

²₂2

  9 o 1      f g   9 o 1  

2

f 1

2

  f 1 

²₂

  

l g 

₂2

l g  f l 

₂2

f l 

2

f 1

2

  f 1 

²₂

  

Claim 1 Claim 1

 Claim 1 Claim 1 : Let : Let f f be linear. be linear.

w.l.o.g., assume w.l.o.g., assume

then

then   global constant global constant c=min{p,1-p} c=min{p,1-p}

s.t.

s.t.     i 2,...,n : f i i 2,...,n : f i             c 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 c c 