Analysis of Boolean Functions Fourier Analysis, Projections, Influence, Junta, Etc

1
Analysis of Boolean Functions Fourier
Analysis,Projections, Influence,Junta,Etc
Slides prepared with help of Ricky Rosen
2
Introduction

• Objectives
• Codes and Juntas using Fourier Analysis.
• Overview
• Codes basic definitions
• Testing Hadamard code
• Testing Long code
• Junta Test

3
Codes and Boolean Functions

• Def an m-bit binary code is a subset of the set
  of all m-binary strings
• C?-1,1m
• The distance of a code C, is the minimum, over
  all pairs of legal-words (in C), of the Hamming
  distance between the two words
• A Boolean function over n binaryvariables, is a
  2n-bit string
• Hence, a set of Boolean functions can be
  considered as a 2n-bits code

4
Hadamard Code

• In the Hadamard code the set of legal-words
  consists of all multiplicative functions.(linear
  if over 0,1) C?S S ? nnamely all
  characters

5
Hadamard Test

• Given a Boolean f, choose random x and y check
  that f(x)f(y)f(xy)
• Prop(perfect completeness) a legal Hadamard word
  (a character) always passes this test

6
Hadamard Test Soundness

• Prop(soundness)
• Proof if f(x)?f(y)f(xy) , then
  f(x)?f(y)?f(xy)1 else f(x)?f(y)?f(xy)-1
• ?x,yf(x)f(y)f(xy)1?Prf(x)f(y)f(xy)
  -1?Prf(x)f(y)??f(xy)??½(1??) -½(1-?) ?

7
proof
8
Proof cont.

• Conclusion
• Proof 1 (probabilistic method) consider
  random variables that with probability are
  valued And its expectation is gt? then one of
  its variables gt ?.
• Proof 2 (algebraic) ifthen
• How large can be?

9
Juntas

• A function is a J-junta if its value depends on
  only J variables.

1
1
1
-1
-1
1
1
1
1
1
-1
-1
-1
-1
1
-1

• A Dictatorship is 1-junta

1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
10
Long-Code

• In the long-code the set of legal-words consists
  of all monotone dictatorships
• This is the most extensive binary code, as its
  bits represent all possible binary values over n
  elements

11
Long-Code

• Encoding an element e?n
• Ee legally-encodes an element e if Ee fe

T
F
F
T
T
12
Testing Long-code

• Def(a long-code list-test) given a code-word f,
  probe it in a constant number of entries, and
• Completeness (not perfect) accept almost always
  if f is a monotone dictatorship
• Soundness reject w.h.p if f does not have a
  sizeable fraction of its Fourier weight
  concentrated on a small set of variables, that
  is, if ?? a semi-Junta J?n s.t.
• Note a long-code list-test, distinguishes
  between the case f is a dictatorship, to the case
  f is far from a junta.

13
Motivation Testing Long-code

• The long-code list-test are essential tools in
  proving hardness results.
• Hence finding simple sufficient-conditions for a
  function to be a junta is important.

14
What about a Hadamar like test?

• completeness?
• yes
• Soundness?
• We would like something like
• Which functions will pass the test?
• all the characters for start
• and many more
• no

15
Perturbation

• Def denote by ?p the distribution over all
  subsets of n, which assigns probability to a
  subset x as follows
• independently, for each i?n, let
• i?x with probability 1-p
• i?x with probability p

16
Long-Code Test

• Given a Boolean f, choose random x and y, and
  choose z??? check that f(x)f(y)f(xyz)
• Prop(completeness) a legal long-code word (a
  dictatorship) passes this test w.p. 1-?

17
Long-code Test Soundness

• Prop(soundness)
• Proof

18
Proof cont.

• Try to find k s.t.
• to get
• k can be determined according to ? and the size
  of the character.

19
List decoding

• This test does not allow list-decoding a
  function.
• Problem the function ?? passes the test, as well
  as functions close to it.
• Solution (?) assume f is folded (odd)
  f(x)-f(-x) for every x.(make sure you
  understand why this is a solution)

20
Junta Test

1. Definitions
2. Independence test
3. The size test
4. Soundness and completeness of the tests

21
Definitions Variation

• Def the variation of f (extension of
  influence)
• Intuition if I is very influential on f then the
  function will go wild on y??PI hence the
  expected variance (?variation) is large.

22
Variation cont.

• Prop the following is an equivalent definitions
  to the variation of f
• Recall

23

• Recall the variance of f
• Hence

24
Proof Cont.

• Recall
• Therefore (by Parseval)

25
High vs Low Frequencies

• Def The section of a function f above k is
• and the low-frequency portion is

26
Junta Test

• Def A Junta test is as follows
• A distribution over l queries
• For each l-tuple, a local-test that either
  accepts or rejects Tx1, , xl 1, -1l?T,F
• s.t. for a j-junta f
• whereas for any f which is not (?, j)-Junta
• The test (l) will be polynomial in j/??

27
Fourier Representation of influence

• Recall consider the I-average function on
  PI
• which in Fourier representation is
• and

28
Subsets Influence

• Recall The Variation of a subset I ? n on a
  Boolean function f is
• and the low-frequency influence

29
Independence-Test

• The I-independence-test on a Boolean function f
  is, for
• Lemma

30
proof
31
(No Transcript)
32
Junta Test

• The junta-size-test JT on a Boolean function f is
• Randomly partition n to I1, .., Ir for rgtgtj2
• Run IT t times on each Ih for tgtgtj2/?
• Accept if no more than j of the Ih fail IT

33
Completeness

• completeness for a j-junta f only those Ih that
  contain a member of the Junta fail IT.
• ?No more than j sets can fail the test.

34
Soundness

• Soundness if f passes the test w.p. ½ then f is
  (?,j) -junta
• Proof utilize bounds on the variation of those
  Ih that pass IT.
• Intuition A set, Ih, has probability of
  ½Variation to fail IT once.If Ih passes IT t
  times, one expects that ½Variation(Ih) lt 1/t

35

• Formally (if you insist)The probability of the
  event that Ih fails IT is p ½Variation.
• The probability of Ih to pass IT t times is
  (1-p)t.
• If it happens w.h.p then
• e-pt gt (1-p)t gt ½
• -pt gt ln(½)
• p lt 1/t(-ln(½)) lt 1/t

36
Soundness Proof

• Assume the premise. Fix ?gt1/t and let
• using the bound on p we prove that if f passes JT
  then f is ? close to a Jjunta.
• Prop if JT succeeds w.p gt ½ then J j
• Proof otherwise,
• J spreads among Ih w.h.p.
• and for any Ih s.t. Ih?J ? ? it must be that
  VariationIh(f) gt ?

37
j spread

• For a random partition, by birthday problem, for
  rgtj2 and fix some j variables from J, w.h.p. no
  two members of J fall in the same Ih.
• Choose r s.t. w.p. ? ¾ at least j1 members of J
  are spread in distinct Ihs.

38

• ?I containing one of the variables in J and a
  fixed i?I
• and since I contains a variable of J,
  variationIgt??
• Since j2/?ltltt we can choose ? s.t.
  ?gt2/tln(j1)ln4
• Now, for a random partition one (like you) can
  bound the probability that one of the Ih that
  contain of of the j1 members of J passes IT t
  times by the union bound
• The probability of the size test to succeed is
  lt ¼ ¾?¼ lt ½
• contradiction to the assumption that the
  test succeeds w.p gt½

J does not spread between j1 Is
J does spreads between j1 Is and IT succeeds
39
Where are we?

• We concluded that if the JT succeeds w.p gt ½ then
  Jltj
• Now what?
• We will show that almost all the weight of f is
  concentrated on J.
• How ?
• (1) Show that the total weight on the high
  frequencies is small.
• (2) Show that the total weight of the low
  frequencies on the characters that are not
  contained in J is small.

40
(1)High Frequencies Contribute Little

• Prop k gtgt r log r implies
• Proof by the Coupon Collector Problem, a
  character S of size larger than k spreads w.h.p.
  (gt¾) over all the Ih (namely, intersects every
  Ih), hence contributes to the influence of all
  parts.For this event

In every Ih ? member of S (S s.t. Sgtk)
41
High frequencies cont.

• Use union bound to bound the probability that one
  of I1 Ij1 to pass IT t times test.
• This probability is lt
• ? w.p. at least ¾? ¾ 9/16 JT fails.
• contradiction

8j2/ ? ltt
J2gtln(j1)ln4
Prob. for spreading over all Ih
Prob. That the first j1 groups fail the size test
42
(2)Almost all Weight is on J

• Lemma
• Proof assume by way of contradiction
  otherwise since
• for a random partition w.h.p. (gt¾) ( by a
  Chernoff like bound (?i? influencei lt?)for
  every h
• however, since for any I
• And also

43

• Similar to the last claim, the probability to
  fail the test in such an event is at least ¾.
• ? the test fails w.p gt ½
• contradiction
• Note for this union bound t200rk/?ln(j1)ln4

44
Find the Close Junta

• Now, since
• consider the (non Boolean)
• which, if rounded outside J

45

• Then
• The distance of f from g --the closest Boolean
  function to g-- is no more than fs
• By the triangle inequality

Q.E.D
46
Juntas

• A function is a J-junta if its value depends on
  only J variables.

1
1
1
-1
-1
1
1
1
1
1
-1
-1
-1
-1
1
-1

• A Dictatorship is 1-junta

1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
47
? - Noise sensitivity
Choose a subset (I) of variables Each var is in
the set with probability ?

• The noise sensitivity of a function f is the
  probability that f changes its value when
  changing a subset of its variables according to
  the ?p distribution.

Flip each value of the subset (I) with
probability p
What is the new value of f?
1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
-1
48
Noise sensitivity and juntas
Choose a subset (I) of variables Each var is in
the set with probability ?
Flip each value of the subset (I) with
probability p
What is the new value of f? W.H.P STAY THE SAME
1
1
1
1
1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
-1

• Juntas are noise insensitive (stable)
• Thm Bourgain Kindler S Noise insensitive
  (stable) Boolean functions are Juntas

49
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

50
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

51
High vs. Low Frequencies

• Def The section of a function f above k is
• and the low-frequency portion is

52
Low-degree B.f are Juntas KS

• 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-?(½)
  Then ? constant ?gt0 s.t. any Boolean function
  fP(n)?-1,1 satisfying is an ?,j-junta for
  jO(?-2k3?2k)

53
Freidgut Theorem

• Thm any Boolean f is an ?, j-junta for
• Proof
• Specify the junta J
• Show the complement of J has little influence

54
Codes and Boolean Functions

• Def an m-bit code is a subset of the set of all
  the m-binary string
• C?-1,1m
• The distance of a code C is the minimum, over all
  pairs of legal-words (in C), of the Hamming
  distance between the two words
• Note A Boolean function over n binary variables
  is a 2n-bit string
• Hence, a set of Boolean functions can be
  considered as a 2n-bits code

55
Long-Code ? Monotone-Dictatorship

• In the long-code, the legal code-words are all
  monotone dictatorships C?i i?
  nnamely, all the singleton characters

56
Open Questions

• Mechanism Design show a non truth-revealing
  protocol in which the pay is smaller (Nash
  equilibrium when all agents tell the truth?)
• Hardness of Approximation
• MAX-CUT
• Coloring a 3-colorable graph with fewest colors
• Graph Properties find sharp-thresholds for
  properties
• Analysis show weakest condition for a function
  to be a Junta
• Apply Concentration of Measure techniques to
  other problems in Complexity Theory

57
(No Transcript)
58
Specify the Junta

• Set k?(as(f)/?), and ?2-?(k)
• Let
• Well prove
• and let
• hence, J is a ?,j-junta, and J2O(k)

59
Functions Vector-Space

• A functions f is a vector
• Addition fg(x) f(x) g(x)
• Multiplication by scalar c?f(x)
  c?f(x)

60
Hadamard Code

• In the Hadamard code theset of legal-words
  consists of all multiplicative (linear if over
  0,1) functions C?S S ? nnamely all
  characters

61
Hadamard Test

• Given a Boolean f, choose random x and y check
  that f(x)f(y)f(xy)
• Prop(completeness) a legal Hadamard word (a
  character) always passes this test

62
Hadamard Test Soundness

• Prop(soundness)
• Proof

63
Testing Long-code

• Def(a long-code list-test) given a code-word f,
  probe it in a constant number of entries, and
• accept almost always if f is a monotone
  dictatorship
• reject w.h.p if f does not have a sizeable
  fraction of its Fourier weight concentrated on a
  small set of variables, that is, if ?? a
  semi-Junta J?n s.t.
• Note a long-code list-test, distinguishes
  between the case f is a dictatorship, to the case
  f is far from a junta.

64
Motivation Testing Long-code

• The long-code list-test are essential tools in
  proving hardness results.
• Hence finding simple sufficient-conditions for a
  function to be a junta is important.

65
High Frequencies Contribute Little

• Prop k gtgt r log r implies
• Proof a character S of size larger than k
  spreads w.h.p. over all parts Ih, hence
  contributes to the influence of all parts.If
  such characters were heavy (gt?/4), then surely
  there would be more than j parts Ih that fail the
  t independence-tests

66
Altogether

• Lemma
• Proof

67
Altogether
68
Beckner/Nelson/Bonami Inequality

• Def let T? be the following operator on any f,
  
• Prop
• Proof

69
Beckner/Nelson/Bonami Inequality

• Def let T? be the following operator on any f,
  
• Thm for any pr and ?((r-1)/(p-1))½

70
Beckner/Nelson/Bonami Corollary

• Corollary 1 for any real f and 2r1
• Corollary 2 for real f and rgt2

71
Perturbation

• Def denote by ?? the distribution over all
  subsets of n, which assigns probability to a
  subset x as follows
• independently, for each i?n, let
• i?x with probability 1-?
• i?x with probability ?

72
Long-Code Test

• Given a Boolean f, choose random x and y, and
  choose z??? check that f(x)f(y)f(xyz)
• Prop(completeness) a legal long-code word (a
  dictatorship) passes this test w.p. 1-?

73
Long-code Tests

• 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

74
Efficient Long-code Tests

• For some applications, it suffices if the test
  may accept illegal code-words, nevertheless, ones
  which have short list-decoding
• Def(a long-code list-test) given a code-word w,
  probe it in 2/3 places, and
• accept w.h.p if w is a monotone dictatorship,
• reject w.h.p if w is not even approximately
  determined by a short 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.

75
General Direction

• These tests may vary
• The long-code list-test a, in particular the
  biased case version, seem essential in proving
  improved hardness results for approximation
  problems
• Other interesting applications
• Hence finding simple, weak as possible,
  sufficient-conditions for a function to be a
  junta is important.