Correlation Decay from Cyclic Polymorphisms Jonah Brown-Cohen 1 - PowerPoint PPT Presentation

Correlation Decay from Cyclic Polymorphisms Jonah Brown-Cohen 1 Prasad Raghavendra 1 1 UC Berkeley Dagstuhl, June 19, 2015 Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Outline Goal: Understand properties of polymorphisms applied to probability distributions on satisfying assignments Distributions on Satisfying Assignments Correlation Decay Applications Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Distributions on Satisfying Assignments Let C be a constraint on two variables X and Y Let µ be a joint distribution on ( X , Y ) satisfying C Helpful to visualize µ as a bipartite graph X Y 1 3 0 0 1 3 1 3 1 1 Figure: Distribution µ for boolean constraint ( X ⇒ Y ) Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Distributions on Satisfying Assignments We say µ has no perfect correlations if the corresponding graph is connected X Y 1 3 0 0 1 3 1 3 1 1 Figure: No perfect correlations Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Distributions on Satisfying Assignments We say µ has no perfect correlations if the corresponding graph is connected X Y 1 2 0 0 1 2 1 1 Figure: Perfect correlation Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Distributions on Satisfying Assignments Let µ X and µ Y be marginal distributions Write µ × = µ X × µ Y for corresponding product distribution X Y 1 3 0 0 1 3 1 3 1 1 Figure: Original distribution µ Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Distributions on Satisfying Assignments Let µ X and µ Y be marginal distributions Write µ × = µ X × µ Y for corresponding product distribution X Y 2 9 0 0 4 9 1 9 2 9 1 1 Figure: Product distribution µ × Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Polymorphisms Acting on Distributions Given a k -ary polymorphism f : D k → D , define a new distribution f ( µ ) as follows: Sample k pairs ( X 1 , Y 1 ) . . . ( X k , Y k ) from µ Output the pair ( f ( X 1 . . . X k ) , f ( Y 1 . . . Y k )) Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Polymorphisms Acting on Distributions Given a k -ary polymorphism f : D k → D , define a new distribution f ( µ ) as follows: Sample k pairs ( X 1 , Y 1 ) . . . ( X k , Y k ) from µ Output the pair ( f ( X 1 . . . X k ) , f ( Y 1 . . . Y k )) Note: f ( µ ) is always a distribution on satisfying assignments f ( µ × ) = f ( µ ) × Can apply f many times to get f ( k ) ( µ ) Question: What happens to the distribution f ( k ) ( µ )? Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Polymorphisms Acting on Distributions Recall that a polymorphism f is cyclic if the output is invariant under all cyclic permutations of the input. Theorem (Correlation Decay) Suppose f is a cyclic polymorphism and µ a distribution on satisfying assignments to some constraint C. Then k →∞ � f ( k ) ( µ ) − f ( k ) ( µ × ) � 1 = 0 lim Intuitively, cyclic polymorphisms jumble-up joint distributions until the two variables eventually become independent. Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? Let µ be distribution given by graph below. X Y 1 4 + ρ 2 0 0 1 4 − ρ 2 1 4 − ρ 2 1 4 + ρ 2 1 1 Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? Let µ be distribution given by graph below. X Y 1 4 + ρ 2 0 0 1 4 − ρ 2 1 4 − ρ 2 1 4 + ρ 2 1 1 Marginals of µ are uniform on { 0 , 1 } Note Pr µ [ X = Y ] = 1 2 + ρ Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? Let f : {− 1 , 1 } 3 → {− 1 , 1 } be the majority operation on three bits. Want to show that f ( µ ) is closer to product distribution than µ . Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? Let f : {− 1 , 1 } 3 → {− 1 , 1 } be the majority operation on three bits. Want to show that f ( µ ) is closer to product distribution than µ . µ [ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2+1 Pr 2 E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? Let f : {− 1 , 1 } 3 → {− 1 , 1 } be the majority operation on three bits. Want to show that f ( µ ) is closer to product distribution than µ . µ [ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2+1 Pr 2 E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] Writing Fourier expansion we have: E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] = Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? Let f : {− 1 , 1 } 3 → {− 1 , 1 } be the majority operation on three bits. Want to show that f ( µ ) is closer to product distribution than µ . µ [ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2+1 Pr 2 E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] Writing Fourier expansion we have:   � f S ˆ ˆ � � E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] = f T E X i Y j  i ∈ S j ∈ T S , T Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? Let f : {− 1 , 1 } 3 → {− 1 , 1 } be the majority operation on three bits. Want to show that f ( µ ) is closer to product distribution than µ . µ [ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2+1 Pr 2 E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] Writing Fourier expansion we have:   � f S ˆ ˆ � � E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] = f T E X i Y j  i ∈ S j ∈ T S , T � ˆ � f 2 = E [ X i Y i ] S S i ∈ S Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? Let f : {− 1 , 1 } 3 → {− 1 , 1 } be the majority operation on three bits. Want to show that f ( µ ) is closer to product distribution than µ . µ [ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2+1 Pr 2 E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] Writing Fourier expansion we have:   � f S ˆ ˆ � � E [ f ( X 1 , X 2 , X 3 ) f ( Y 1 , Y 2 , Y 3 )] = f T E X i Y j  i ∈ S j ∈ T S , T � ˆ � f 2 = E [ X i Y i ] S S i ∈ S � f 2 ˆ S (2 ρ ) | S | = S Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? So we know: Pr[ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2 + 1 � ˆ f 2 S (2 ρ ) | S | 2 S Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? So we know: Pr[ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2 + 1 � ˆ f 2 S (2 ρ ) | S | 2 S If only non-zero Fourier coefficients have | S | = 1 then nothing changes: 1 2 + 1 2(2 ρ ) = 1 2 + ρ Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? So we know: Pr[ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2 + 1 � ˆ f 2 S (2 ρ ) | S | 2 S If only non-zero Fourier coefficients have | S | = 1 then nothing changes: 1 2 + 1 2(2 ρ ) = 1 2 + ρ | S | =1 ˆ f 2 But for majority � S < 1 − c for a constant c > 0: Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? So we know: Pr[ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2 + 1 � ˆ f 2 S (2 ρ ) | S | 2 S If only non-zero Fourier coefficients have | S | = 1 then nothing changes: 2 + 1 1 2(2 ρ ) = 1 2 + ρ | S | =1 ˆ f 2 But for majority � S < 1 − c for a constant c > 0: S (2 ρ ) | S | ≤ (1 − c )(2 ρ ) + c (2 ρ ) 2 < 2 ρ � f 2 ˆ S Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Why Should Correlation Decay? So we know: Pr[ f ( X 1 , X 2 , X 3 ) = f ( Y 1 , Y 2 , Y 3 )] = 1 2 + 1 � ˆ f 2 S (2 ρ ) | S | 2 S If only non-zero Fourier coefficients have | S | = 1 then nothing changes: 1 2 + 1 2(2 ρ ) = 1 2 + ρ | S | =1 ˆ f 2 But for majority � S < 1 − c for a constant c > 0: S (2 ρ ) | S | ≤ (1 − c )(2 ρ ) + c (2 ρ ) 2 < 2 ρ � f 2 ˆ S So for ( X , Y ) sampled from f ( µ ), Pr[ X = Y ] < 1 2 + ρ Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms

Correlation Rough idea of the proof: show that correlation of µ goes down each time we apply f . Definition (Correlation) For a joint distribution µ the correlation is given by ρ ( µ ) := sup µ [ f ( X ) g ( Y )] E f , g where the supremum is over functions f , g with E [ f ] = E [ g ] = 0 and E [ f 2 ] = E [ g 2 ] = 1. Two extreme cases to keep in mind: When X , Y independent: E [ f ( X ) g ( Y )] = E [ f ( X )] E [ g ( Y )] = 0 When X = Y always: can take f = g to obtain E [ f ( X ) g ( Y )] = E [ f ( X ) 2 ] = 1 Jonah Brown-Cohen, Prasad Raghavendra Correlation Decay from Cyclic Polymorphisms