harmonic analysis of boolean functionshatami/comp760-2011/lecture0.pdf · scribing two lectures: 15...
TRANSCRIPT
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Harmonic Analysis of Boolean Functions
Lectures: MW 10:05-11:25 in MC 103Office Hours: MC 328, by appointment([email protected]).Evaluation:
Assignments: 60 %Presenting a paper at the end of the term: 20 %Scribing two lectures: 15 %Attending lectures: 5 %
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Tentative plan
Overview:Basic functional analysis.Basic Fourier analysis of discrete Abelian groups.
Mathematical theory:Influences, Noise operator; Discrete Log-Sobolovinequalities, Hyper-contractivity, Threshold Phenomena,Noise sensitivity, etc.
Applications to computer science:Property testing.Machine Learning.Circuit Complexity.Communication complexity.
![Page 3: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/3.jpg)
Tentative plan
Overview:Basic functional analysis.Basic Fourier analysis of discrete Abelian groups.
Mathematical theory:Influences, Noise operator; Discrete Log-Sobolovinequalities, Hyper-contractivity, Threshold Phenomena,Noise sensitivity, etc.
Applications to computer science:Property testing.Machine Learning.Circuit Complexity.Communication complexity.
![Page 4: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/4.jpg)
Tentative plan
Overview:Basic functional analysis.Basic Fourier analysis of discrete Abelian groups.
Mathematical theory:Influences, Noise operator; Discrete Log-Sobolovinequalities, Hyper-contractivity, Threshold Phenomena,Noise sensitivity, etc.
Applications to computer science:Property testing.Machine Learning.Circuit Complexity.Communication complexity.
![Page 5: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/5.jpg)
What are we going to study?
Boolean Functions
f : {0,1}n → {0,1}.
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What is Harmonic Analysis of Boolean Functions?
Harmonic AnalysisFocuses on the quantitative properties of functions, and howthese quantitative properties change when apply various (oftenquite explicit) operators.
Fourier analysis
Studies functions by decomposing them intoa linear combination of “symmetric” functions.These symmetric functions are usually explicit,and are often associated with physical conceptssuch as frequency or energy.
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What is Harmonic Analysis of Boolean Functions?
Harmonic AnalysisFocuses on the quantitative properties of functions, and howthese quantitative properties change when apply various (oftenquite explicit) operators.
Fourier analysis
Studies functions by decomposing them intoa linear combination of “symmetric” functions.These symmetric functions are usually explicit,and are often associated with physical conceptssuch as frequency or energy.
![Page 8: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/8.jpg)
Examples I: CircuitComplexity
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Circuit complexity
QuestionWhat can one say about the Fourier expansion of functionscomputable with small circuits (with gates ∧,∨,¬)?
Theorem (Linial, Mansour, Nisan 1993)The Fourier expansion of every such functions is alwaysconcentrated on low frequencies.
Corollary: Parity cannot be computed with a small circuit.
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Circuit complexity
QuestionWhat can one say about the Fourier expansion of functionscomputable with small circuits (with gates ∧,∨,¬)?
Theorem (Linial, Mansour, Nisan 1993)The Fourier expansion of every such functions is alwaysconcentrated on low frequencies.
Corollary: Parity cannot be computed with a small circuit.
![Page 11: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/11.jpg)
Circuit complexity
QuestionWhat can one say about the Fourier expansion of functionscomputable with small circuits (with gates ∧,∨,¬)?
Theorem (Linial, Mansour, Nisan 1993)The Fourier expansion of every such functions is alwaysconcentrated on low frequencies.
Corollary: Parity cannot be computed with a small circuit.
![Page 12: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/12.jpg)
Examples II: Influences
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We study Boolean functions f : {0,1}n → {0,1}.
f
x1
x2
xn
...f(x)
Think of them as voting systems.
...
Two candidates 0 and 1.Everybody votes 0 or 1.f determines the winner.
![Page 14: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/14.jpg)
We study Boolean functions f : {0,1}n → {0,1}.
f
x1
x2
xn
...f(x)
Think of them as voting systems.
...
Two candidates 0 and 1.Everybody votes 0 or 1.f determines the winner.
![Page 15: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/15.jpg)
We study Boolean functions f : {0,1}n → {0,1}.
f
x1
x2
xn
...f(x)
Think of them as voting systems.
...
Two candidates 0 and 1.Everybody votes 0 or 1.f determines the winner.
![Page 16: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/16.jpg)
We study Boolean functions f : {0,1}n → {0,1}.
f
x1
x2
xn
...f(x)
Think of them as voting systems.
...
Two candidates 0 and 1.
Everybody votes 0 or 1.f determines the winner.
![Page 17: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/17.jpg)
We study Boolean functions f : {0,1}n → {0,1}.
f
x1
x2
xn
...f(x)
Think of them as voting systems.
...
Two candidates 0 and 1.Everybody votes 0 or 1.
f determines the winner.
![Page 18: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/18.jpg)
We study Boolean functions f : {0,1}n → {0,1}.
f
x1
x2
xn
...f(x)
Think of them as voting systems.
...
Two candidates 0 and 1.Everybody votes 0 or 1.f determines the winner.
![Page 19: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/19.jpg)
Consider a function f : {0,1}n → {0,1} and a voter i .
...
How important is the vote of the i-th person?(Find a mathematical definition of the influence of a variable.)
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Consider a function f : {0,1}n → {0,1} and a voter i .
...
How important is the vote of the i-th person?(Find a mathematical definition of the influence of a variable.)
![Page 21: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/21.jpg)
Consider a function f : {0,1}n → {0,1} and a voter i .
...
How important is the vote of the i-th person?(Find a mathematical definition of the influence of a variable.)
x1 x2 x3 f0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 11 1 0 11 1 1 1
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Consider a function f : {0,1}n → {0,1} and a voter i .
...
How important is the vote of the i-th person?(Find a mathematical definition of the influence of a variable.)
x1 x2 x3 f0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 11 1 0 11 1 1 1
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Consider a function f : {0,1}n → {0,1} and a voter i .
...
How important is the vote of the i-th person?(Find a mathematical definition of the influence of a variable.)
x1 x2 x3 f0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 11 1 0 11 1 1 1
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Consider a function f : {0,1}n → {0,1} and a voter i .
...
How important is the vote of the i-th person?(Find a mathematical definition of the influence of a variable.)
x1 x2 x3 f0 0 0 00 0 1 00 1 0 00 1 1 01 0 0 01 0 1 11 1 0 11 1 1 1
Ii =scenarios that xi mattered
2n
I1 = 68
I3 = 28
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Definition (In Probabilistic Language:)
Consider f : {0,1}n → {0,1}, and the i-th person.
Everybody else votes uniformly at random.Ii = Pr[ i-th voter can change the outcome].
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Definition (In Probabilistic Language:)
Consider f : {0,1}n → {0,1}, and the i-th person.Everybody else votes uniformly at random.
Ii = Pr[ i-th voter can change the outcome].
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Definition (In Probabilistic Language:)
Consider f : {0,1}n → {0,1}, and the i-th person.Everybody else votes uniformly at random.Ii = Pr[ i-th voter can change the outcome].
![Page 28: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/28.jpg)
Questionf : {0,1}n → {0,1} with minimum total influence?
AnswerThe constant function f = 0 or f = 1. The total influence is 0.
QuestionBalanced f : {0,1}n → {0,1} with minimum total influence?
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Questionf : {0,1}n → {0,1} with minimum total influence?
AnswerThe constant function f = 0 or f = 1. The total influence is 0.
QuestionBalanced f : {0,1}n → {0,1} with minimum total influence?
![Page 30: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/30.jpg)
Questionf : {0,1}n → {0,1} with minimum total influence?
AnswerThe constant function f = 0 or f = 1. The total influence is 0.
QuestionBalanced f : {0,1}n → {0,1} with minimum total influence?
![Page 31: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/31.jpg)
Example (Dictatorship)
...
f (x) = x1 or f (x) = 1− x1.
I1 = 1, I2 = I3 = . . . = In = 0.
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Example (Dictatorship)
...
f (x) = x1 or f (x) = 1− x1.
I1 = 1, I2 = I3 = . . . = In = 0.
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Example (Dictatorship)
...
f (x) = x1 or f (x) = 1− x1.
I1 = 1, I2 = I3 = . . . = In = 0.
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More examples ofInfluences
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Example (Majority)
f (x) = Majority(x1, . . . , xn).
I1 = I2 = . . . = In ≈ 1√n .
Example (Parity)
f (x) = x1 ⊕ x2 ⊕ . . .⊕ xn.
I1 = I2 = . . . = In = 1.
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Example (Majority)
f (x) = Majority(x1, . . . , xn).
I1 = I2 = . . . = In ≈ 1√n .
Example (Parity)
f (x) = x1 ⊕ x2 ⊕ . . .⊕ xn.
I1 = I2 = . . . = In = 1.
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Example (Majority)
f (x) = Majority(x1, . . . , xn).
I1 = I2 = . . . = In ≈ 1√n .
Example (Parity)
f (x) = x1 ⊕ x2 ⊕ . . .⊕ xn.
I1 = I2 = . . . = In = 1.
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Example (Majority)
f (x) = Majority(x1, . . . , xn).
I1 = I2 = . . . = In ≈ 1√n .
Example (Parity)
f (x) = x1 ⊕ x2 ⊕ . . .⊕ xn.
I1 = I2 = . . . = In = 1.
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Total influence
Dictator:∑
Ii = 1 + 0 + . . .+ 0 = 1.
Parity:∑
Ii = 1 + . . .+ 1 = n.Majority:
∑Ii ≈ 1√
n + . . . 1√n =√
n.
QuestionWhich functions have constant O(1) total influence?
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Total influence
Dictator:∑
Ii = 1 + 0 + . . .+ 0 = 1.Parity:
∑Ii = 1 + . . .+ 1 = n.
Majority:∑
Ii ≈ 1√n + . . . 1√
n =√
n.
QuestionWhich functions have constant O(1) total influence?
![Page 41: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/41.jpg)
Total influence
Dictator:∑
Ii = 1 + 0 + . . .+ 0 = 1.Parity:
∑Ii = 1 + . . .+ 1 = n.
Majority:∑
Ii ≈ 1√n + . . . 1√
n =√
n.
QuestionWhich functions have constant O(1) total influence?
![Page 42: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/42.jpg)
Total influence
Dictator:∑
Ii = 1 + 0 + . . .+ 0 = 1.Parity:
∑Ii = 1 + . . .+ 1 = n.
Majority:∑
Ii ≈ 1√n + . . . 1√
n =√
n.
QuestionWhich functions have constant O(1) total influence?
![Page 43: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/43.jpg)
Junta session one week after the 1973 coup in Chile.
Definition (Junta)
There is a small set of voters {i1, . . . , ik} who decide theelection
f (x) := g(xi1 , . . . , xik ).
![Page 44: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/44.jpg)
Definition (Junta-Recall)
There is a small set of voters {i1, . . . , ik} who decide theelection
f (x) := g(xi1 , . . . , xik ).
Everybody outside the junta has influence 0.∑Ii ≤ k .
Friedgut: The inverse is essentially true.
Theorem (Friedgut’98)
If the total influence is constant then f is approximately a junta.
Proof is based on the proof of the KKL inequality.
![Page 45: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/45.jpg)
Definition (Junta-Recall)
There is a small set of voters {i1, . . . , ik} who decide theelection
f (x) := g(xi1 , . . . , xik ).
Everybody outside the junta has influence 0.
∑Ii ≤ k .
Friedgut: The inverse is essentially true.
Theorem (Friedgut’98)
If the total influence is constant then f is approximately a junta.
Proof is based on the proof of the KKL inequality.
![Page 46: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/46.jpg)
Definition (Junta-Recall)
There is a small set of voters {i1, . . . , ik} who decide theelection
f (x) := g(xi1 , . . . , xik ).
Everybody outside the junta has influence 0.∑Ii ≤ k .
Friedgut: The inverse is essentially true.
Theorem (Friedgut’98)
If the total influence is constant then f is approximately a junta.
Proof is based on the proof of the KKL inequality.
![Page 47: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/47.jpg)
Definition (Junta-Recall)
There is a small set of voters {i1, . . . , ik} who decide theelection
f (x) := g(xi1 , . . . , xik ).
Everybody outside the junta has influence 0.∑Ii ≤ k .
Friedgut: The inverse is essentially true.
Theorem (Friedgut’98)
If the total influence is constant then f is approximately a junta.
Proof is based on the proof of the KKL inequality.
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Definition (Junta-Recall)
There is a small set of voters {i1, . . . , ik} who decide theelection
f (x) := g(xi1 , . . . , xik ).
Everybody outside the junta has influence 0.∑Ii ≤ k .
Friedgut: The inverse is essentially true.
Theorem (Friedgut’98)
If the total influence is constant then f is approximately a junta.
Proof is based on the proof of the KKL inequality.
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Theorem (KKL inequality 1988)Let f be a balanced function. Then
maxi
Ii &log n
n.
It has many applications in computer science.The proof was influential.It is based on hyper-contractivity, a phenomenon inharmonic analysis.It introduced this tool to the community of computerscience and combinatorics.
![Page 50: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/50.jpg)
Theorem (KKL inequality 1988)Let f be a balanced function. Then
maxi
Ii &log n
n.
It has many applications in computer science.
The proof was influential.It is based on hyper-contractivity, a phenomenon inharmonic analysis.It introduced this tool to the community of computerscience and combinatorics.
![Page 51: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/51.jpg)
Theorem (KKL inequality 1988)Let f be a balanced function. Then
maxi
Ii &log n
n.
It has many applications in computer science.The proof was influential.
It is based on hyper-contractivity, a phenomenon inharmonic analysis.It introduced this tool to the community of computerscience and combinatorics.
![Page 52: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/52.jpg)
Theorem (KKL inequality 1988)Let f be a balanced function. Then
maxi
Ii &log n
n.
It has many applications in computer science.The proof was influential.It is based on hyper-contractivity, a phenomenon inharmonic analysis.
It introduced this tool to the community of computerscience and combinatorics.
![Page 53: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/53.jpg)
Theorem (KKL inequality 1988)Let f be a balanced function. Then
maxi
Ii &log n
n.
It has many applications in computer science.The proof was influential.It is based on hyper-contractivity, a phenomenon inharmonic analysis.It introduced this tool to the community of computerscience and combinatorics.
![Page 54: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/54.jpg)
Example III: PhaseTransitions
![Page 55: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/55.jpg)
Erdös-Rényi graph
In early sixties Erdös and Rényi invented the notion of arandom graph G(n,p):
A random graph on n vertices, whereevery edge is present independently with probability p.
G(n, p)
e
Pr[e ∈ G(n, p)] = p
![Page 56: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/56.jpg)
Erdös-Rényi graph
In early sixties Erdös and Rényi invented the notion of arandom graph G(n,p):A random graph on n vertices, where
every edge is present independently with probability p.
G(n, p)
e
Pr[e ∈ G(n, p)] = p
![Page 57: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/57.jpg)
Erdös-Rényi graph
In early sixties Erdös and Rényi invented the notion of arandom graph G(n,p):A random graph on n vertices, whereevery edge is present independently with probability p.
G(n, p)
e
Pr[e ∈ G(n, p)] = p
![Page 58: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/58.jpg)
Thresholds
They observed that some fundamental graph properties suchas connectivity exhibit a threshold as p increases.
1
1
p
Pr[G(n, p) is connected]
![Page 59: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/59.jpg)
Thresholds
This is an instance of the phenomenon of phase transition instatistical physics which explains the rapid change of behaviorin many physical processes.
![Page 60: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/60.jpg)
The critical probability
DefinitionLet f : {0,1}n → {0,1} be an increasing function. The criticalprobability pc is defined as
Prx∼µpc
[f (x) = 1] = 1/2.
1
p
Pr[G(n, p) is connected]
pc =lnnn
![Page 61: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/61.jpg)
Theorem (Bollobás-Thomason 1987)
Let f : {0,1}n → {0,1} be increasing. Then
Prx∼µp
[f (x) = 1] ={
o(1) p � pc1− o(1) p � pc .
1
p
Pr[f(x) = 1]
pc
transition interval
![Page 62: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/62.jpg)
Theorem (Bollobás-Thomason 1987)
Let f : {0,1}n → {0,1} be increasing. Then
Prx∼µp
[f (x) = 1] ={
o(1) p � pc1− o(1) p � pc .
1
p
Pr[f(x) = 1]
pc
transition interval
![Page 63: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/63.jpg)
Theorem (Bollobás-Thomason 1987)
Let f : {0,1}n → {0,1} be increasing. Then
Prx∼µp
[f (x) = 1] ={
o(1) p � pc1− o(1) p � pc .
1
p
Pr[f(x) = 1]
pc
transition interval
![Page 64: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/64.jpg)
ExampleConnectivity:
1
p
Pr[G(n, p) is connected]
pc =lnnn
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ExampleContaining a triangle:
1
p
Pr[triangle]
pc
![Page 66: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/66.jpg)
sharpness of threshold
One of the main questions that arises in studying phasetransitions is:
“How sharp is the threshold?”
That is how short is the interval in which the transitionoccurs.
1
p
Pr[f(x) = 1]
pc
transition interval
![Page 67: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/67.jpg)
sharpness of threshold
One of the main questions that arises in studying phasetransitions is:
“How sharp is the threshold?”That is how short is the interval in which the transitionoccurs.
1
p
Pr[f(x) = 1]
pc
transition interval
![Page 68: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/68.jpg)
Theorem (Recall Bollobás-Thomason)
Every increasing function f : {0,1}n → {0,1} exhibits athreshold:
Prx∼µp
[f (x) = 1] ={
o(1) p � pc1− o(1) p � pc .
Definition (Sharp threshold)
An increasing function f : {0,1}n → {0,1} exhibits a sharpthreshold, if for all ε > 0,
Prx∼µp
[f (x) = 1] ={
o(1) p ≤ (1− ε)pc1− o(1) p ≥ (1 + ε)pc .
![Page 69: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/69.jpg)
Theorem (Recall Bollobás-Thomason)
Every increasing function f : {0,1}n → {0,1} exhibits athreshold:
Prx∼µp
[f (x) = 1] ={
o(1) p � pc1− o(1) p � pc .
Definition (Sharp threshold)
An increasing function f : {0,1}n → {0,1} exhibits a sharpthreshold, if for all ε > 0,
Prx∼µp
[f (x) = 1] ={
o(1) p ≤ (1− ε)pc1− o(1) p ≥ (1 + ε)pc .
![Page 70: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/70.jpg)
ExampleContaining a triangle does not exhibit a sharp threshold.
1
p
Pr[triangle]
pc
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ExampleConnectivity exhibits a sharp threshold.
1
p
Pr[G(n, p) is connected]
pc =lnnn
![Page 72: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/72.jpg)
ExampleWhat about more complicated properties such as
Satisfiability of a 3-SAT formula.3-colorability of a graph.
Is there a general approach to such questions?
QuestionWhat can we say about f : {0,1}n → {0,1} if it does not exhibita sharp threshold?
![Page 73: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/73.jpg)
ExampleWhat about more complicated properties such as
Satisfiability of a 3-SAT formula.3-colorability of a graph.
Is there a general approach to such questions?
QuestionWhat can we say about f : {0,1}n → {0,1} if it does not exhibita sharp threshold?
![Page 74: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/74.jpg)
ExampleWhat about more complicated properties such as
Satisfiability of a 3-SAT formula.3-colorability of a graph.
Is there a general approach to such questions?
QuestionWhat can we say about f : {0,1}n → {0,1} if it does not exhibita sharp threshold?
![Page 75: Harmonic Analysis of Boolean Functionshatami/comp760-2011/lecture0.pdf · Scribing two lectures: 15 % Attending lectures: 5 %. Tentative plan Overview: Basic functional analysis](https://reader033.vdocuments.us/reader033/viewer/2022050113/5f4a118b91bb81620f672980/html5/thumbnails/75.jpg)
ExampleWhat about more complicated properties such as
Satisfiability of a 3-SAT formula.3-colorability of a graph.
Is there a general approach to such questions?
QuestionWhat can we say about f : {0,1}n → {0,1} if it does not exhibita sharp threshold?