Boolean function analysis: beyond the hypercube Yuval Filmus - PowerPoint PPT Presentation

Boolean function analysis: beyond the hypercube Yuval Filmus Technion — Israel Institute of Technology CAALM’19

What is Boolean function analysis? Dimension-independent properties of functions { 0 , 1 } n → { 0 , 1 } Many applications to combinatorics and computational complexity

Motivating example: Erd˝ os–Ko–Rado theorem � [ n ] � Suppose F ⊂ is intersecting, k = pn , p < 1 / 2. k

Motivating example: Erd˝ os–Ko–Rado theorem � [ n ] � Suppose F ⊂ is intersecting, k = pn , p < 1 / 2. k � n − 1 1 |F| ≤ � . k − 1

Motivating example: Erd˝ os–Ko–Rado theorem � [ n ] � Suppose F ⊂ is intersecting, k = pn , p < 1 / 2. k � n − 1 1 |F| ≤ � . k − 1 � n − 1 2 |F| = � = ⇒ F is a star, i.e. { A : i ∈ A } . k − 1

Motivating example: Erd˝ os–Ko–Rado theorem � [ n ] � Suppose F ⊂ is intersecting, k = pn , p < 1 / 2. k � n − 1 1 |F| ≤ � . k − 1 � n − 1 2 |F| = � = ⇒ F is a star, i.e. { A : i ∈ A } . k − 1 � n − 1 3 |F| ≈ � = ⇒ F ≈ a star. k − 1

Motivating example: Erd˝ os–Ko–Rado theorem � [ n ] � Suppose F ⊂ is intersecting, k = pn , p < 1 / 2. k � n − 1 1 |F| ≤ � . k − 1 Lov´ asz: spectral proof using theta function. � n − 1 2 |F| = � = ⇒ F is a star, i.e. { A : i ∈ A } . k − 1 � n − 1 3 |F| ≈ � = ⇒ F ≈ a star. k − 1

Motivating example: Erd˝ os–Ko–Rado theorem � [ n ] � Suppose F ⊂ is intersecting, k = pn , p < 1 / 2. k � n − 1 1 |F| ≤ � . k − 1 Lov´ asz: spectral proof using theta function. � n − 1 2 |F| = � = ⇒ F is a star, i.e. { A : i ∈ A } . k − 1 Boolean degree 1 function is a dictator. � n − 1 3 |F| ≈ � = ⇒ F ≈ a star. k − 1

Motivating example: Erd˝ os–Ko–Rado theorem � [ n ] � Suppose F ⊂ is intersecting, k = pn , p < 1 / 2. k � n − 1 1 |F| ≤ � . k − 1 Lov´ asz: spectral proof using theta function. � n − 1 2 |F| = � = ⇒ F is a star, i.e. { A : i ∈ A } . k − 1 Boolean degree 1 function is a dictator. � n − 1 3 |F| ≈ � = ⇒ F ≈ a star. k − 1 Boolean almost degree 1 function is almost a dictator.

Classical Boolean function analysis Fundamental theorem Every function f : {± 1 } n → R has unique expansion as multilinear polynomial, the Fourier expansion : � ˆ � f ( x 1 , . . . , x n ) = f ( S ) x S , where x S = x i . i ∈ S S ⊆ [ n ]

Classical Boolean function analysis Fundamental theorem Every function f : {± 1 } n → R has unique expansion as multilinear polynomial, the Fourier expansion : � ˆ � f ( x 1 , . . . , x n ) = f ( S ) x S , where x S = x i . i ∈ S S ⊆ [ n ] Degree of f = degree of Fourier expansion.

Classical Boolean function analysis Fundamental theorem Every function f : {± 1 } n → R has unique expansion as multilinear polynomial, the Fourier expansion : � ˆ � f ( x 1 , . . . , x n ) = f ( S ) x S , where x S = x i . i ∈ S S ⊆ [ n ] Degree of f = degree of Fourier expansion. Dictator: function depending on one coordinate. d -Junta: function depending on d coordinates. deg f ≤ d iff f is linear combination of d -juntas.

Boolean degree 1 functions Question Suppose f : {± 1 } n → {± 1 } has degree 1. What does f look like?

Boolean degree 1 functions Question Suppose f : {± 1 } n → {± 1 } has degree 1. What does f look like? deg f ≤ 1 ⇐ ⇒ f ( x 1 , . . . , x n ) = a 0 + a 1 x 1 + · · · + a n x n .

Boolean degree 1 functions Question Suppose f : {± 1 } n → {± 1 } has degree 1. What does f look like? deg f ≤ 1 ⇐ ⇒ f ( x 1 , . . . , x n ) = a 0 + a 1 x 1 + · · · + a n x n . Dictator theorem If f : {± 1 } n → {± 1 } has degree 1 then f ∈ {± 1 , ± x 1 , . . . , ± x n } .

Boolean almost degree 1 functions Refined question Suppose f : {± 1 } n → {± 1 } satisfies x ∼{± 1 } n [( f ( x ) − g ( x )) 2 ] = ǫ E for some g : {± 1 } n → R of degree 1. What does f look like?

Boolean almost degree 1 functions Refined question Suppose f : {± 1 } n → {± 1 } satisfies x ∼{± 1 } n [( f ( x ) − g ( x )) 2 ] = ǫ E for some g : {± 1 } n → R of degree 1. What does f look like? Friedgut–Kalai–Naor (FKN) theorem Suppose f : {± 1 } n → {± 1 } satisfies � f > 1 � 2 = ǫ . Then Pr[ f � = h ] = O ( ǫ ) for some h ∈ {± 1 , ± x 1 , . . . , ± x n } .

Boolean function analysis on the slice The slice or Johnson scheme is � n � � [ n ] � ( x 1 , . . . , x n ) ∈ { 0 , 1 } n : � = x i = k . k i =1

Boolean function analysis on the slice The slice or Johnson scheme is � n � � [ n ] � ( x 1 , . . . , x n ) ∈ { 0 , 1 } n : � = x i = k . k i =1 Fundamental theorem (Dunkl) � [ n ] � Every function f : → R has unique expansion as multilinear k polynomial P of degree ≤ min( k , n − k ) such that n ∂ P � = 0 . ∂ x i i =1 Examples: 1 , ( x 1 − x 2 ) , ( x 1 − x 2 )( x 3 − x 4 ) , . . .

Degree of functions on the slice Fundamental theorem (Dunkl) � [ n ] � Every function f : → R has unique expansion as multilinear k polynomial P of degree ≤ min( k , n − k ) such that n ∂ P � = 0 . ∂ x i i =1

Degree of functions on the slice Fundamental theorem (Dunkl) � [ n ] � Every function f : → R has unique expansion as multilinear k polynomial P of degree ≤ min( k , n − k ) such that n ∂ P � = 0 . ∂ x i i =1 Degree of f = degree of unique expansion.

Degree of functions on the slice Fundamental theorem (Dunkl) � [ n ] � Every function f : → R has unique expansion as multilinear k polynomial P of degree ≤ min( k , n − k ) such that n ∂ P � = 0 . ∂ x i i =1 Degree of f = degree of unique expansion. Dictator: function depending on one coordinate. d -Junta: function depending on d coordinates. deg f ≤ d iff f is linear combination of d -juntas.

Degree of functions on the slice Fundamental theorem (Dunkl) � [ n ] � Every function f : → R has unique expansion as multilinear k polynomial P of degree ≤ min( k , n − k ) such that n ∂ P � = 0 . ∂ x i i =1 Degree of f = degree of unique expansion. Dictator theorem holds (except for trivial cases). FKN theorem holds for 0 ≪ k / n ≪ 1.

Erd˝ os–Ko–Rado theorem Spectral argument of Lov´ asz Let k = pn , p < 1 / 2. � [ n ] � If F ⊂ is intersecting and F is not too small then k � n − 1 �� 1 − C � 1 > 1 F � 2 � |F| ≤ . k − 1

Erd˝ os–Ko–Rado theorem Spectral argument of Lov´ asz Let k = pn , p < 1 / 2. � [ n ] � If F ⊂ is intersecting and F is not too small then k � n − 1 �� 1 − C � 1 > 1 F � 2 � |F| ≤ . k − 1 Corollaries � n − 1 1 |F| ≤ � . k − 1

Erd˝ os–Ko–Rado theorem Spectral argument of Lov´ asz Let k = pn , p < 1 / 2. � [ n ] � If F ⊂ is intersecting and F is not too small then k � n − 1 �� 1 − C � 1 > 1 F � 2 � |F| ≤ . k − 1 Corollaries � n − 1 1 |F| ≤ � . k − 1 � n − 1 2 |F| = � = ⇒ deg 1 F = 1. k − 1 Dictator theorem: F is a star.

Erd˝ os–Ko–Rado theorem Spectral argument of Lov´ asz Let k = pn , p < 1 / 2. � [ n ] � If F ⊂ is intersecting and F is not too small then k � n − 1 �� 1 − C � 1 > 1 F � 2 � |F| ≤ . k − 1 Corollaries � n − 1 1 |F| ≤ � . k − 1 � n − 1 2 |F| = � = ⇒ deg 1 F = 1. k − 1 Dictator theorem: F is a star. F � 2 = O ( ǫ ). � n − 1 3 |F| = (1 − ǫ ) ⇒ � 1 > 1 � = k − 1 FKN theorem: F is O ( ǫ )-close to a star.

FKN theorem for small k ? Let p := k / n = o (1) and ǫ ≫ p 2 . � [ n ] � Consider g : → R defined as k g := x 1 + · · · + x √ ǫ/ p

FKN theorem for small k ? Let p := k / n = o (1) and ǫ ≫ p 2 . � [ n ] � Consider g : → R defined as k g := x 1 + · · · + x √ ǫ/ p ∼ Bin ( √ ǫ/ p , p )

FKN theorem for small k ? Let p := k / n = o (1) and ǫ ≫ p 2 . � [ n ] � Consider g : → R defined as k g := x 1 + · · · + x √ ǫ/ p ∼ Bin ( √ ǫ/ p , p ) ∼ Poisson ( √ ǫ )

FKN theorem for small k ? Let p := k / n = o (1) and ǫ ≫ p 2 . � [ n ] � Consider g : → R defined as k g := x 1 + · · · + x √ ǫ/ p ∼ Bin ( √ ǫ/ p , p ) ∼ Poisson ( √ ǫ ) This shows that Pr[ g = 0] ≈ 1 − √ ǫ . Pr[ g = 1] ≈ √ ǫ − ǫ . Pr[ g ≥ 2] ≈ ǫ .

FKN theorem for small k ? Let p := k / n = o (1) and ǫ ≫ p 2 . � [ n ] � Consider g : → R defined as k g := x 1 + · · · + x √ ǫ/ p ∼ Bin ( √ ǫ/ p , p ) ∼ Poisson ( √ ǫ ) This shows that Pr[ g = 0] ≈ 1 − √ ǫ . Pr[ g = 1] ≈ √ ǫ − ǫ . Pr[ g ≥ 2] ≈ ǫ . Therefore O ( ǫ ) ≈ f := x 1 ∨ · · · ∨ x √ ǫ/ p g

FKN theorem for small k FKN theorem on the slice (F.) Let p := k / n ≤ 1 / 2. → { 0 , 1 } satisfies � f > 1 � 2 = ǫ then either f or 1 − f is � [ n ] � If f : k O ( ǫ )-close to a disjunction of m variables, where � √ ǫ � �� m = max 1 , O . p