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 [BL, KKL] [BL, KKL] and and

related issues, has been introduced to related issues, has been introduced to

tackle

tackle social choice social choice problems, problems,

furthermore has motivated a magnificent furthermore has motivated a magnificent

sequence of works, related to economics sequence of works, related to economics

[K], percolation [BKS], Hardness of [K], percolation [BKS], Hardness of

approximation [DS]

approximation [DS]

Revolving around the

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

analysis of Boolean functions … …

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, and (electronic) envelope, and

have the system decided, have the system decided,

not necessarily according to not necessarily according to

majority…

majority…

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

Florida wing- has the power Florida wing- has the power

to flip some votes to flip some votes

Power

Power

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

 

 It turns out It turns out : 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 very few of unless taking into account only very few of

the votes

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   [n] [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~U

n

f x f' x

Pr ^{          }

x~U

n

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

Long-Code Long-Code

 In the long-code In the long-code L:[n] L:[n]   {0,1} {0,1} ^{2 2 n n} each element is each element is encoded by an

encoded by an 2 2 ^{n n} -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 _{e e} legally-encodes legally-encodes an element an element e e if if E E _{e e} = f = f _{e e}

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 list-decoding:

which have short 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 2/3 places, and

probe it in 2/3 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 , that is, if

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

between the case w w is a is a dictatorship dictatorship , to the , to the

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 (Kindler&Safra) Thm (Kindler&Safra) : a Boolean function : a Boolean function f f with small with small high-frequency weight

high-frequency 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 [BL,KKL,F] [BL,KKL,F]

high-frequency weight

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

[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~   _{p p} ^{I I} 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

   

[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] x [n]

z II

Noise-Sensitivity

Noise-Sensitivity

Fourier/Walsh Transform Fourier/Walsh Transform

Write

Write f:{-1, 1} f:{-1, 1} ^{n n}   {-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 ) )

?????

?????

Make sense now to consider the degree of

Make sense now to consider the degree of f f or or to break it according to the various degrees of to break it according to the various degrees of

the monomials..

the monomials..

High/Low Frequencies and High/Low Frequencies and

their Weights their Weights

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

 

f S 1 

 ^{f S 1 }

²

^{  }

 ^

 

^{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???? : :

 Corollary Corollary : : _{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 

  

 

  

 

  ^

    ^  

x P I S I

E g x g S

   

  

  

    ^  

x P I S I

E g x g S

   

  

  

Variation Variation

Def Def : the : the variation variation of of f f : :

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

definitions to the variation variation of of 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

Low-freq Variation and Low-freq Variation and

Low-freq Average-Sensitivity Low-freq Average-Sensitivity

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

  ^f   ^{f f (S) S 2}

 ^k    ^k   ^

as ^{ k}   ^{f }  variation ^{ k}   ^{f }  ^{f (S) S  2}

as variation

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 

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, and (electronic) envelope, and

have the system decided, have the system decided,

not necessarily according to not necessarily according to

majority…

majority…

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

Florida wing- has the power Florida wing- has the power

to flip some votes to flip some votes

Power Power

Of course they’ll have to discuss it over

dinner….

First Attempt:

First Attempt:

Following Freidgut’s Proof Following Freidgut’s Proof

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  

P([n])

Following Freidgut - Cont Following Freidgut - Cont

Lemma Lemma : :

Proof Proof : :

Now, lets bound each argument:

Now, lets bound each argument:

Prop Prop : : 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

P([n])

J

k 2

k f 

^{k 2}2

k f 

^ 2

 

²

^{ }

as f ^{ }   f S S ^

²

^{ } 

S

as f   f S S ^ 

   

    ^{ }

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)

f

f(S) 2 f(S)

2 f(S) 2 f(S)

2 f 2 as f

f ^



   

 

 

 

  

   

    



   

   



 

 

 

 

 variation   vari on ati i  

    ^{ }

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)

f

f(S) 2 f(S)

2 f(S) 2 f(S)

2 f 2 as f

f ^



   

 

 

 

  

   

    

    



   

   



 

 

 

 

 variation vari on ati i

Following Freidgut - Cont Following Freidgut - Cont

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)

In order for those to hit close to 1 or -1 most of In order for those to hit close to 1 or -1 most of

the time, they must avoid the law of large the time, they must avoid the law of large

numbers, namely be almost entirely placed on numbers, namely be almost entirely placed on

one singleton [by Chernoff like bound]

one singleton [by Chernoff like bound]

Thm[FKN, ext.]: Assume

Thm[FKN, ext.]: Assume f f is close to is close to linear, linear, then

then f f is is close to close to shallow shallow ( (   a constant a constant

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 1 ₁ ,…,I ,…,I r _r , for , for r >> k r >> k

 w.h.p w.h.p f f _{I I} [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

  ^{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  

P([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 either a constant-function is either a constant-function or a dictatorship, for any

or a dictatorship, for any x x Still,

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

dictatorship for every x x , hence the , hence the

P([n])

J I

₁

I

₂

I

_r

I

almost linear almost linear

  almost shallow almost shallow

Theorem(

Theorem( [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

Dictatorship and its Singleton Dictatorship and its Singleton

 Prop Prop : if : if f f _{I I} [x] [x] is a dictatorship, then is a dictatorship, then

  coordinate coordinate i i s.t. s.t. (where (where p p is the is the bias).

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

 



I

   

f x i ^

I

       p f x i  p

    

h i       p h i  p

 

²2

{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

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,

  ^{      }

 

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} ₂ ⁴ ₂ ^{k 2} ₂

x~ Pr g x M g M g

 ^{ }

_m

^      ^   ₁  ^{ 4k  k} ₂ ⁴  ₂ ^{ k 2} ₂

x~ Pr g x M g M g

 ^    ^    ^{ }  ^

 Lemma Lemma : :

 Proof Proof : let : let , 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 : : ^{i 2} _max ^{max 2 max}

i I

a 1 a a a



  

 ^{i 2} _max ^{max 2 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

     ^  ^

Obtaining the Lemma Obtaining the Lemma

 It remains to show that indeed: It remains to show that indeed:

 Prop1 Prop1 : :

 Prop2 Prop2 : :

 

^I

 ^{ } 

I f  E x~ _   I f x I   variation I   f  E x~ _

^I

  variation I  f x I ^{ }    variation variation

   

J1 r

4k k 2

I~ I 2

E _

1^Jr

^  variation   f    O   ^  f ^ 

4k k 2

I~ I 2

E _ ^  variation f    O  ^  f ^

 

    

I

2 2

x~ I

S I S I

E _ f x S f S

 

   

 

  

  ^{    }   ^{  }

I

2 2

x~ I

S I S I

E _ f x S f S

 

   

 

  

    ^

• Recall Recall

• However However

  _{   } ^{ }  

I

f

x P I

E

y P I

var f x y

I

 

   

      

variation  

  ^{ }    

I

f

x P I

E

y P I

var f x y

I

 

   

      

variation

    _{ } ^{   }

I

f x

I 0 y P I

var f x y

I 0



 

  

variation

I

 f x

I

 

0



y P I

var f x y _{ }

I

^{ }

0

^{ }



 

  

variation

Obtaining the Lemma – Cont.

Obtaining the Lemma – Cont.

 Prop3 Prop3 : :

 Proof Proof : separate by freq: : separate by freq:

 Small freq: Small freq:

 Large freq: Large freq:

 Corollary Corollary (from props 2,3): (from props 2,3):

   

J I

1r

2 k 2

I~ ,x~ 2

S I 1

E _{ } f S O f ^

 

 

  

 

  ^{  }   

J I

1r

2 k 2

I~ ,x~ 2

S I 1

E _{ } f S O f ^

 

 

  

 

  ^ 

   

J I

1r

2 2 2

I~ ,x~

S k S I 1 I

S k

E f S Pr S I 1 | k r O 1

r

k 1

 

 

 

          

  

 

 

 ^{  }  ^{ }

J I

1r

2 2 2

I~ ,x~

S k S I 1 I

S k

E f S Pr S I 1 | k r O 1

r k 1

 

 

 

          

  

 

 

 ^ 

 

J I

1r

2 k 2

I~ ,x~ 2

S I 1 S k

E

_{ }

f S f

^

 

 

  

 

 

 

 ^{  }

J I

1r

2 k 2

I~ ,x~ 2

S I 1 S k

E

_{ }

f S f

^

 

 

  

 

 

 

 ^

 

2 2

 ²     ²

  

Obtaining the Lemma – Cont.

Obtaining the Lemma – Cont.

 Recall: by corollary from [FKN], Recall: by corollary from [FKN], Either

Either or or

 Hence Hence

 By By Corollary Corollary

 Combined with Combined with Prop1 Prop1 we obtain: we obtain:

   

f i       p f i   p

  n   f  M f ^{ 1 2} 2

variation _{  n}   ^{f  M f  1 2} ₂

variation

      ^{ }

J I J I J I

1r 1r 1r

1 2

I I I I

I~ ,x~ I~ ,x~ I~ ,x~ 2

E

_{ }

f x Pr x D E

_{ }

M f x

 







    

 variation          ^{ }  

J I J I J I

1r 1r 1r

1 2

I I I I

I~ ,x~ I~ ,x~ I~ ,x~ 2

E

_{ }

f x Pr x D E

_{ }

M f x

 







    

 variation     

|D |D

_II

| | is small is small

   

J I

1r

2 2

1 k

I~ ,x~ I 2 2

E

_^J1 _^I

   f x  

^

    O f  

^

r

2 2

1 k

I~ ,x~ I 2 2

E

_{ }

   f x

^

    O f

^

   

J1 r

4k k 2

I~ I 2

E ^I~ _

^1J_r

^  variation ^I   f    O   ^{ 4k}  f ^{ k 2 2} 

E _ ^  variation f    O  ^  f ^

Important Lemma Important 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

Beckner/Nelson/Bonami Inequality Beckner/Nelson/Bonami 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 f f s.t. s.t. f f ^{>k >k} =0 =0

   

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

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.

 

_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

g ⁴  ^{ 4k} g 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

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

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           0 0   _{   } i i

₀₀

 

f f     

f f    f f i f f i             0 0   _{   } i i

₀₀

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 00 ₀₀ =z =z , , x x 10 ₁₀ =z =z   {1} {1} , , x x 01 ₀₁ =z =z   {2} {2} , , x x 11 ₁₁ =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 _{ } i

₀

f f       f i      0  _{ } i

₀

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

?

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

      0 _{ } i

₀

f f       f i      0  _{ } i

₀

f f     f i  

f f       

f f    f f i f f i             0 0   _{   } i i

₀₀

       

    ^{ }   ^{ }

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

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

1

-1 0  

f   

f  f i f 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 _{0 0} =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  ₂ 2   3 o 1      f g   3 o 1  

     

0

f    f i 

  f      f i     

0

 

   

2

f g  ² ₂ 2   9 o 1      f g   9 o 1  

2

f 1

2

  f 1 

²₂

  

l g  ₂ 2

l g  f l  ₂ 2

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  

weight

Characters Each of weight no more than

Each of weight no more than c c 