1. Noise-Insensitive Boolean-Functions are Juntas
Guy Kindler & Muli Safra

2. Dictatorship
Def: a boolean function P([n]){-1,1} is a monotone e-dictatorships --denoted fe--if:

3. We would tend to omit p
Juntas
Def: a boolean function f:P([n]){-1,1} is a j-Junta if J[n] where |J|≤ j, s.t. for every x[n]: f(x) = f(x  J)
Def: f is an [, j]-Junta if  j-Junta f’ s.t.
Def: f is an [, j, p]-Junta if  j-Junta f’ s.t.

4. Codes and Boolean Functions
Def: a code is a mapping of a set of n elements (log n bits’ string) to a set of m-bits strings C:[n]{0,1}m, i.e. C(e) = a1…am
Def: Let Sj={e[n] | C(e)j=T}
Let ={Sj}j[m]
C(1)
C(2) C(3)
C(n)
FF T TF F
TT F
TT T
S1={2,3,n}
Sm={1,n}

5. Codes and Boolean Functions
Def: Let Ee be the encoding of element e.
Consider {Ee}e[n] Each Ee’s truth-table represents a legal--code-word of C ( since C(e) = Ee(S1)…Ee(Sm) )
C(1)
C(2) C(3)
C(n)
FF0 T TF F
TT F
TT T
S1={2,3,n}
Sm={1,n}

6. Long-Code
In the long-code L:[n] {0,1}2n each element is encoded by an 2n-bits
This is the most extensive code, as  = P([n]), i.e. the bits represent all subsets in P([n])

7. Long-Code Encoding an element e[n]:
Ee legally-encodes an element e if Ee = fe F F T T T

8. Motivation – Testing Long-code
Def (a long-code test): given a code-word w, probe it in a constant number of entries, and
accept w.h.p if w is a monotone dictatorship
reject w.h.p if w is not close to any monotone dictatorship

9. Motivation – Testing Long-code
Def(a long-code list-test): given a code-word w, probe it in a constant number of entries, and
accept w.h.p if w is a monotone dictatorship,
reject w.h.p if w is not even approximately determined by a small list of domain elements, that is, if  a Junta J[n] s.t. f is close to f’ and f’(x)=f’(xJ) for all x
Note: a long-code list-test, distinguishes between the case w is a dictatorship, to the case w is far from a junta.

10. Motivation – Testing Long-code
The long-code test, and the long-code list-test are essential tools in proving hardness results. Examples …
Hence finding simple sufficient-conditions for a function to be a junta is important.

11. Background
Thm (Friedgut): a boolean function f with small average-sensitivity is an [,j]-junta
Thm (Bourgain): a boolean function f with small high-frequency weight is an [,j]-junta
Thm (Kindler&Safra): a boolean function f with small high-frequency weight in a p-biased measure is an [,j]-junta
Corollary: a boolean function f with small noise-sensitivity is an [,j]-junta
Parameters: average-sensitivity, high-frequency weight, noise-sensitivity

12. Noise-Sensitivity
Idea: check how the value of f changes when the input is changed not on one, but on several coordinates.
[n] I z x

13. Noise-Sensitivity
Def(,p,x[n] ): Let 0<<1, and xP([n]). Then y~,p,x, if y = (x\I) z where
I~[n] is a noise subset, and
z~ pI is a replacement.
Def(-noise-sensitivity): let 0<<1, then
Note: deletes a coordinate in x w.p. (1-p), adds a coordinate to x w.p. p. Hence, when p=1/2: equivalent to flipping each coordinate in x w.p. /2.
[n] x I z

14. Noise-Sensitivity – Cont.
Advantage: very efficiently testable (using only two queries) by a perturbation-test.
Def (perturbation-test): choose x~p, and y~,p,x, check whether f(x)=f(y). The success is proportional to the noise-sensitivity of f.
Prop: the -noise-sensitivity is given by

15. Relation between Parameters
Prop: small ns small high-freq weight
Proof: therefore: if ns is small, then Hence the high frequencies must have small weights (as ).
Prop: small as small high-freq weight
Proof:

16. Average and Restriction
Def: Let I[n], xP([n]\I), the restriction function is
Def: the average function is
Note: I y x
[n] I y y y y y x

17. Fourier Expansion
Prop:
Corollary:

18. Variation Def: the variation of f (formerly called influence):
Prop: the following are equivalent definitions to the variation of f:

19. Proof
Recall
Therefore

20. Proof – Cont.
Recall
Therefore (by Parseval):

21. High/Low Frequencies and their Weights
Def: the high-frequency portion of f:
Def: the low-frequency portion of f:
Def: the high-frequency-weight is:
Def: the low-frequency-weight is:

22. Low-freq variation and Low-freq average-sensitivity
Def: the low-frequency variation is:
Def: the average sensitivity is
And in Fourier representation:
Def: the low-frequency average sensitivity is:

23. Main Results
Theorem:  constant >0 s.t. any boolean function f:P([n]){-1,1} satisfying is an [,j]-junta for j=O(-2k32k).
Corollary: fix a p-biased distribution p over P([n]). Let >0 be any parameter. Set k=log1-(1/2). Then  constant >0 s.t. any boolean function f:P([n]){-1,1} satisfying is an [,j]-junta for j=O(-2k32k).

24. First Attempt: Following Freidgut’s Proof
Thm: any boolean function f is an [,j]-junta for
Proof:
Specify the junta where, let k=O(as(f)/) and fix =2-O(k)
Show the complement of J has small variation
P([n]) J

25. Following Freidgut - Cont
P([n]) J
Following Freidgut - Cont
Lemma:
Proof:
Now, lets bound each argument:
Prop:
Proof: characters of sizek contribute to the average-sensitivity at least (since )

26. Following Freidgut - Cont
True only since this is a {-1,0,1} function. So we cannot proceed this way with only ask! 
we do not know whether as(f) is small! 
Following Freidgut - Cont
Prop:
Proof:

27. Important Lemma
Lemma: >0, s.t. for any  and any function g:P([m]) , the following holds:
Low-freq
high-freq

28. Beckner/Nelson/Bonami Inequality
Def: let T be the following operator on any f,
Thm: for any p≥r and ≤((r-1)/(p-1))½
Corollary: for f s.t. f>k=0

29. Probability Concentration
Simple Bound:
Proof:
Low-freq Bound: Let g:P([m])  s.t. g>k=0, let >0, then >0 s.t.
Proof: recall the corollary: 

30. Lemma’s Proof Now, let’s prove the lemma:
Bounding low and high freq separately: ,
Low-freq bound
simple bound

31. Shallow Function
Def: a function f is linear, if only singletons have non-zero weight
Def: a function f is shallow, if f is either a constant or a dictatorship.
Claim: boolean linear functions are shallow.
weight
Character size
k n

32. Boolean Linear  Shallow
Claim: boolean linear functions are shallow.
Proof: let f be boolean linear function, we next show:
{io} s.t. (i.e. )
And conclude, that either or i.e. f is shallow

33. Claim 1 Claim 1: let f be boolean linear function, then {io} s.t.
Proof: w.l.o.g assume
for any z{3,…,n}, consider x00=z, x10=z{1}, x01=z{2}, x11=z{1,2}
then
Next value must be far from {-1,1},
A contradiction! (boolean function)
Therefore 1 -1 ?

34. Claim 2 Claim 2: let f be boolean function, s.t. Then either or
Proof: consider f() and f(i0):
Then
but f is boolean, hence
therefore 1 -1

35. Proving FKN: almost-linear  close to shallow
Theorem: Let f:P([n])  be linear,
Let
let i0 be the index s.t is maximal
then
Note: f is linear, hence w.l.o.g., assume i0=1, then all we need to show is: We show that in the following claim and lemma.

36. Corollary
Corollary: Let f be linear, and then  a shallow boolean function g s.t.
Proof: let , let g be the boolean function closest to l. Then, this is true, as
is small (by theorem),
and additionally is small, since

37. Each of weight no more than c
Claim 1
Claim 1: Let f be linear. w.l.o.g., assume then global constant c=min{p,1-p} s.t.
{} {1} {2} {i} {n} {1,2} {1,3} {n-1,n} S {1,..,n}
weight
Characters
Each of weight no more than c

38. Proof of Claim1 Proof: assume
for any z{3,…,n}, consider x00=z, x10=z{1}, x01=z{2}, x11=z{1,2}
then
Next value must be far from {-1,1} !
A contradiction! (to ) 1 -1 ?

39. Lemma Lemma: Let g be linear, let assume , then
note
Lemma: Let g be linear, let assume , then
Corrolary: The theorem follows from the combination of claim1 and the lemma:
Let m be the minimal index s.t.
Consider
If m=2: the theorem is obtained (by lemma)
Otherwise -- a contradiction to minimality of m :

40. Lemma’s Proof Lemma’s Proof: Note Hence, all we need to show is that
Intuition:
Note that |g| and |b| are far from 0 (since |g| is -close to 1, and c-close to b).
Assume b>0, then for almost all inputs x, g(x)=|g(x)| (as )
Hence E[g]  E[|g(x)|], and
therefore var(g)  var(|g|)

41. E2[g] - E2[|g|] = 2E2[|g|1{f<0}]  o() (by Azuma’s inequality)
Proof-map:
|g|,|b| are far from 0
g(x)=|g(x)| for almost all x
E[g]  E[|g|]
var(g)  var(|g|)   
E2[g] - E2[|g|] = 2E2[|g|1{f<0}]  o() (by Azuma’s inequality)
We next show var(g)  var(|g|):
By the premise
however
therefore 

42. Central Ideas: Linear Functions and Random Partition
Idea 1: recall
(theorem’s premise)
Assume f is close to linear then f is close to shallow (by [FKN]).
Idea 2: Let .
Partition J into I1,…,Ir.
r is large, hence w.h.p fI[x] is close to linear (low freq characters intersect each I by 1 element).
P([n]) I2 Ir I I1 J

43. Variation Lemma Lemma(variation): >0, and r>>k s.t.
P([n])
Variation Lemma I2 Ir I I1
Lemma(variation): >0, and r>>k s.t.
Corollary: for most I and x, fI[x] is almost constant J

44. Using Idea2 By union bound on I1,…,Ir:
P([n])
Using Idea2 I2 Ir I I1
By union bound on I1,…,Ir:
(set )
Let f’(x) = sign( AJ[f](xJ) ). f’ is the boolean function closest to AJ[f], therefore
Hence f is an [,j]-junta. J

45. variation-Lemma - Proof Plan
Lemma(variation): >0, and r>>k s.t.
Sketch for proving the variation lemma:
w.h.p fI[x] is almost linear
w.h.p fI[x] is close to shallow
fI[x] cannot be close to dictatorship too often.

46. Proof-map:
w.h.p fI[x] is almost linear
w.h.p fI[x] is close to shallow
fI[x] cannot be close to dictatorship too often.
Lemma Proof
The low frequencies characters are small, r is rather large, hence w.h.p the characters are linear at each I.
P([n]) I2 Ir I I1 J

47. almost linear  almost shallow
Proof-map:
w.h.p fI[x] is almost linear
w.h.p fI[x] is close to shallow
fI[x] cannot be close to dictatorship too often.
almost linear  almost shallow
Theorem([FKN]): global constant M, s.t. boolean function f, shallow boolean function g, s.t.
Hence, ||fI[x]>1||2 is small  fI[x] is close to shallow!

48. Preliminary Lemma and Props
Prop: if fI[x] is a dictatorship, then coordinate i s.t. (where p is the bias).
Corollary (from [FKN]): global constant M, s.t. boolean function h, either or
weight
Total weight of no more than 1-p
Characters
{1} {2} {i} {n} {1,2} {1,3} {n-1,n} S {1,..,n}

49. Proof-map:
w.h.p fI[x] is almost linear
w.h.p fI[x] is close to shallow
fI[x] cannot be close to dictatorship too often.
Few Dictatorships
Lemma: >0, s.t. for any  and any function g:P([m]) , the following holds:
Def: Let DI be the set of xP(I), s.t. fI[x] is a dictatorship, i.e.
Next we show, that |DI| must be small, hence for most x, fI[x] is constant.

50. |DI| must be small Lemma: Proof: let , then Prev lemma Parseval
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!)

51. Simple Prop
Prop: let {ai}iI be sub-distribution, that is, iIai1, 0ai, then iIai2maxiI{ai}.
Proof:
max n ai
no more than 1 1 n ai
1/amax 1

52. |DI| must be small - Cont
Therefore (since ),
Hence

53. Obtaining the Lemma It remains to show that indeed: Prop1: Prop2:
Recall
However
{S}S are orthonormal, and

54. Obtaining the Lemma – Cont.
Prop3:
Proof: separate by freq:
Small freq:
Large freq:
Corollary(from props 2,3):

55. Obtaining the Lemma – Cont.
Recall: by corollary from [FKN], Either or
Hence
By Corollary
Combined with Prop1 we obtain:
|DI| is small

56. The End

57. XOR Test
Let  be a random procedure for choosing two disjoint subsets x,y s.t.: i[n], ix\y w.p 1/3, iy\x w.p 1/3, and ixy w.p 1/3.
Def(XOR-Test): Pick <x,y>~,
Accept if f(x)f(y),
Reject otherwise.

58. Example
Claim: Let f be a dictatorship, then f passes the XOR-test w.p. 2/3.
Proof: Let i be the dictator, then Pr<x,y>~[f(x)f(y)]=Pr<x,y>~ [ixy]=2/3
Claim: Let f’ be a -close to a dictatorship f, then f’ passes the XOR-test w.p. 2/3 – 2/3(-2).
Proof: see next slide…

60. Local Maximality
Def: f is locally maximal with respect to a test, if f’ obtained from f by a change on one input x0, that is, Pr<x,y>~[f(x)f(y)]  Pr<x,y>~[f’(x)f’(y)]
Def: Let x be the distribution of all y such that <x,y>~.
Claim: if f is locally maximal then f(x) = -sign(Ey~(x)[f(y)]).

61. Claim:
Proof: immediate from the Fourier-expansion, and the fact that yx=

62. Conjecture: Let f be locally maximal (with respect to the XOR-test), assume f passes the XOR-test w.p  1/2 + , for some constant >0, then f is close to a junta.