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NoiseInsensitive BooleanFunctions are Juntas

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Title: NoiseInsensitive BooleanFunctions are Juntas


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
Juntas
We would tend to omit p
  • Def a boolean function fP(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
    Cn?0,1m, i.e. C(e) a1am
  • Def Let Sje?n C(e)jT
  • Let ??Sjj?m

C(1) C(2) C(3) C(n)
FF TTF F TT F TT T
Sm1,n
S12,3,n
5
Codes and Boolean Functions
FF0 TTF F TT F TT T
C(1) C(2) C(3) C(n)
  • Def Let Ee be the encodingof element e.
  • Consider Eee?nEach Ees truth-table
    represents a legal--code-word of C( since C(e)
    Ee(S1)Ee(Sm) )

Sm1,n
S12,3,n
6
Long-Code
  • In the long-code Ln? 0,12n 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

T
F
F
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 (KindlerSafra) 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,xn ) Let 0lt?lt1, 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 0lt?lt1, then
  • Note deletes a coordinate in x w.p.
    ?(1-p), adds a coordinate to x w.p. ?p. Hence,
    when p1/2 equivalent to flipping each
    coordinate in x w.p. ?/2.

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
n
  • 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
  • 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
Proof
  • First, lets show

22
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

23
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

24
Biased Walsh Product
  • Def In the p-biased product distribution ?p, the
    probability of a subset x is
  • The usual Fourier basis ?is not orthogonal with
    respect to the biased inner-product,
  • Hence, we use the Biased Walsh Product

25
Main Results
  • Theorem ? constant ?gt0 s.t. any boolean
    function fP(n)?-1,1 satisfying is an
    ?,j-junta for jO(?-2k3?2k).
  • Corollary fix a p-biased distribution ?p over
    P(n). Let ?gt0 be any parameter. Set
    klog1-?(1/2). Then ? constant ?gt0 s.t. any
    boolean function fP(n)?-1,1 satisfying is
    an ?,j-junta for jO(?-2k3?2k).

26
First Attempt Following Freidguts Proof
  • Thm any boolean function f is an ?,j-junta for
  • Proof
  • Specify the juntawhere, let kO(as(f)/?) and fix
    ?2-O(k)
  • Show the complement of J has small variation

P(n)
J
27
Proving J has small variation suffices
  • Prop Let f be a boolean function, s.t.
    variationJ(f)?? ?, then f is an ?,J-junta.
  • Proof define a junta f as follows f(x)f(x?J)
    then f is a J-junta, andhence

28
Following Freidgut - Cont
  • Lemma
  • Proof
  • Now, lets bound each argument
  • Prop
  • Proof characters of size??k contribute to the
    average-sensitivity at least (since )

29
Following Freidgut - Cont
  • Lemma
  • Proof
  • Now, lets bound each argument
  • Prop
  • Proof characters of size??k contribute to the
    average-sensitivity at least (since )

30
Following Freidgut - Cont
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 as?k! ?
  • Prop
  • Proof

31
Whats Next
  • Preliminaries
  • Important lemma
  • FKN
  • Theorems proof

32
Important Lemma
  • Lemma ??gt0, s.t. for any ? and any function
    gP(m)?? ?, the following holds

high-freq
Low-freq
33
Beckner/Nelson/Bonami Inequality
  • Def let T? be the following operator on any f,
  • Thm for any pr and ?((r-1)/(p-1))½
  • Corollary for f s.t. fgtk0

34
Beckner/Nelson/Bonami Corollary
  • Proof

35
Probability Concentration
  • Simple Bound
  • Proof
  • Low-freq Bound Let gP(m)?? ? s.t. ggtk0, let
    ?gt0, then ??gt0 s.t.
  • Proof recall the corollary

?
36
Lemmas Proof
  • Now, lets prove the lemma
  • Bounding low and high freq separately???,

simple bound
Low-freq bound
37
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
Charactersize
0 1 2 3 k n
38
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

39
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 x00z, x10z?1,
    x01z?2, x11z?1,2
  • then .
  • Next value must be far from -1,1,
  • A contradiction! (boolean function)
  • Therefore

?
40
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 x00z, x10z?1,
    x01z?2, x11z?1,2
  • then .
  • But this is impossible as f(x00),f(x10) ,f(x01),
    f(x11) ? -1,1, hence their distances cannot all
    be gt0 !
  • Therefore .

?
41
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

42
Linearity and Dictatorship
  • Prop Let f be a balanced linear boolean function
    then f is a dictatorship.
  • Prooff(?),f(i0)??-1,1, hence
  • Prop Let f be a balanced boolean function s.t.
    as(f)1, then f is a dictatorship.
  • Proof , but f is balanced, (i.e. ),
    therefore f is also linear.

43
Proving FKN almost-linear ? close to shallow
  • Theorem Let fP(n)?? ? be linear,
  • Let
  • let i0 be the index s.t. is maximal
  • then
  • Note f is linear, hence w.l.o.g., assume
    i01, then all we need to show is We show
    that in the following claim and lemma.

44
Corollary
  • Corollary Let f be linear, andthen ? 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

45
Claim 1
  • Claim 1 Let f be linear. w.l.o.g., assumethen
    ?global constant cminp,1-p s.t.

46
Proof of Claim1
  • Proof assume
  • for any z?3,,n, consider x00z, x10z?1,
    x01z?2, x11z?1,2
  • then
  • Next value must be far from -1,1 !
  • A contradiction! (to )

?
47
Proof of Claim1
they cannot all be near -1,1!
  • Proof assume .
  • for any z?3,,n, consider x00z, x10z?1,
    x01z?2, x11z?1,2
  • then .
  • Hence
  • Therefore, for a random x this holds w.p. at
    least c, and therefore -- a contradiction.

?
48
Lemma
  • 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 m2 the theorem is obtained (by lemma)
  • Otherwise -- a contradiction to minimality of m

49
Lemmas Proof
  • Lemmas 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 bgt0, then for almost all inputs x,
    g(x)g(x) (as )
  • Hence Eg ? Eg(x), and
  • therefore var(g) ? var(g)

50
Proof-map g,b are far from 0 g(x)g(x) for
almost all x Eg ? Eg var(g) ? var(g)
?
?
?
  • E2g - E2g 2E2g1flt0 ? o(?) (by
    Azumas inequality)
  • We next show var(g) ? var(g)
  • By the premise
  • however
  • therefore

?
51
Central Ideas Linear Functions and Random
Partition
  • Idea 1 recall
  • (theorems 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 fIx is close to linear
    (low freq characters intersect each I by ?1
    element).

P(n)
I2
Ir
I
I1
J
52
Variation Lemma
P(n)
I2
Ir
I
I1
  • Lemma(variation) ??gt0, and rgtgtk s.t.
  • Corollary for most I and x, fIx is almost
    constant

J
53
Using Idea2
P(n)
I2
Ir
I
I1
  • By union bound on I1,,Ir

  • (set )
  • Let f(x) sign( AJf(x?J) ). f is the
    boolean function closest to AJf, therefore
  • Hence f is an ?,j-junta.

J
54
variation-Lemma - Proof Plan
  • Lemma(variation) ??gt0, and rgtgtk s.t.
  • Sketch for proving the variation lemma
  • w.h.p fIx is almost linear
  • w.h.p fIx is close to shallow
  • fIx cannot be close to dictatorship too often.

55
Lemma Proof
  • Proof-map
  • w.h.p fIx is almost linear
  • w.h.p fIx is close to shallow
  • fIx cannot be close to dictatorship too often.
  • 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
56
almost linear ? almost shallow
  • Proof-map
  • w.h.p fIx is almost linear
  • w.h.p fIx is close to shallow
  • fIx cannot be close to dictatorship too often.
  • Theorem(FKN) ?global constant M, s.t.
    ?boolean function f, ?shallow boolean function
    g, s.t.
  • Hence, fIxgt12 is small ?? fIx is close to
    shallow!

57
Preliminary Lemma and Props
  • Prop if fIx is a dictatorship, then
    ?coordinate i s.t. (where p is the
    bias).
  • Corollary (from FKN) ?global constant M, s.t.
    ?boolean function h, eitheror

weight
Total weight of no more than 1-p
Characters
1 2 i n 1,2 1,3 n-1,n S 1,..,n
58
Few Dictatorships
  • Proof-map
  • w.h.p fIx is almost linear
  • w.h.p fIx is close to shallow
  • fIx cannot be close to dictatorship too often.
  • Lemma ??gt0, s.t. for any ? and any function
    gP(m)?? ?, the following holds
  • Def Let DI be the set of x?P(I), s.t. fIx is
    a dictatorship, i.e.
  • Next we show, that DI must be small, hence for
    most x, fIx is constant.

59
DI must be small
Parseval
Prev lemma
  • Lemma
  • Proof let , then

Each S is counted only for one index i?I.
(Otherwise, if S was counted for both i and j in
I, then S?Igt1!)
60
Simple Prop
  • Prop let aii?I be sub-distribution, that is,
    ?i?Iai?1, 0?ai, then ?i?Iai2?maxi?Iai.
  • Proof

61
DI must be small - Cont
  • Therefore(since ),
  • Hence

62
Obtaining the Lemma
  • It remains to show that indeed
  • Prop1
  • Prop2

63
Obtaining the Lemma Cont.
  • Prop3
  • Proof separate by freq
  • Small freq
  • Large freq
  • Corollary(from props 2,3)

64
Obtaining the Lemma Cont.
  • Recall by corollary from FKN, Either or
  • Hence
  • By Corollary
  • Combined with Prop1 we obtain

DI is small
65
prop1
DI must be small
prop2
66
  • The End

67
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, andi?x?y w.p 1/3.
  • Def(XOR-Test) Pick ltx,ygt?,
  • Accept if f(x)??f(y),
  • Reject otherwise.

68
Example
  • Claim Let f be a dictatorship, then f passes the
    XOR-test w.p. 2/3.
  • Proof Let i be the dictator, then
    Prltx,ygt?f(x)??f(y)Prltx,ygt? i?x?y2/3
  • 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

69
(No Transcript)
70
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, Prltx,ygt?f(x)??f(y) ?
    Prltx,ygt?f(x)??f(y)
  • Def Let ?x be the distribution of all y such
    that ltx,ygt?.
  • Claim if f is locally maximal then f(x)
    -sign(Ey?(x)f(y)).

71
  • Claim
  • Proof immediate from the Fourier-expansion, and
    the fact that y?x?

72
  • 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 ?gt0,
    then f is close to a junta.
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