Noise Stability is Low-Dimensional Anindya De, Elchanan Mossel, Joe - PowerPoint PPT Presentation

Noise Stability is Low-Dimensional Anindya De, Elchanan Mossel, Joe Neeman

Gaussian noise stability Take X and Y a pair of ρ -correlated Gaussians in R n (0 < ρ < 1). For a partition A = ( A 1 , . . . , A k ) of R n into k parts, define Stab ρ ( A ) = Pr( X and Y land in the same part) 1

Gaussian noise stability Noise stable Not noise stable 2

Gaussian noise stability Theorem (Borell ’85) For a partition of R n into two parts of equal Gaussian measure, 2 + sin − 1 ρ Stab ρ ( A ) ≤ 1 . π Equality is attained for a partition into half-spaces. 3

Gaussian noise stability Theorem (Borell ’85) For a partition of R n into two parts of equal Gaussian measure, 2 + sin − 1 ρ Stab ρ ( A ) ≤ 1 . π Equality is attained for a partition into half-spaces. • well-known links to computational complexity (KKMO ’05) 3

Gaussian noise stability Theorem (Borell ’85) For a partition of R n into two parts of equal Gaussian measure, 2 + sin − 1 ρ Stab ρ ( A ) ≤ 1 . π Equality is attained for a partition into half-spaces. • well-known links to computational complexity (KKMO ’05) • one-dimensional phenomenon 3

Gaussian noise stability Theorem (???) For a partition of R n into three parts of equal Gaussian measure, ??? 3

Gaussian noise stability Conjecture (Peace sign conjecture) The optimal partition looks like a peace sign. 4

Gaussian noise stability Conjecture (Peace sign conjecture) The optimal partition looks like a peace sign. (two-dimensional phenomenon) 4

Gaussian noise stability Conjecture (Peace sign conjecture) The optimal partition looks like a peace sign. (two-dimensional phenomenon) Conjecture (Multi-dimensional peace sign conjecture) For partitions into k equal measures, the optimal partition occurs in R k − 1 . It looks like a multi-dimensional peace sign. 4

Gaussian noise stability Theorem (De-Mossel-N.) For any k and any ǫ > 0 , there is a computable n 0 = n 0 ( k , ǫ ) such that an ǫ -approximately optimal partition occurs in R n 0 . 5

Gaussian noise stability Theorem (De-Mossel-N.) For any k and any ǫ > 0 , there is a computable n 0 = n 0 ( k , ǫ ) such that an ǫ -approximately optimal partition occurs in R n 0 . Corollary The optimal value of k-part noise stability is computable. 5

Gaussian noise stability Theorem (De-Mossel-N.) For any k and any ǫ > 0 , there is a computable n 0 = n 0 ( k , ǫ ) such that an ǫ -approximately optimal partition occurs in R n 0 . Corollary The optimal value of k-part noise stability is computable. Corollary (sort of) The non-interactive correlation distillation value with k-ary outputs is computable. 5

Correlation distillation Goal: produce uniform output, agree with maximal probability. What is the probability of agreement? 6

Correlation distillation Goal: produce uniform output, agree with maximal probability. What is the probability of agreement? Ghazi-Kamath-Sudan ’16: reduction to correlated Gaussian signals 6

The main theorem Theorem (De-Mossel-N.) For any k and any ǫ > 0 , there is a computable n 0 = n 0 ( k , ǫ ) such that an ǫ -approximately optimal partition occurs in R n 0 . 7

Proof outline Idea: take an optimal partition in R n ( n huge) and try to “simulate” it in R n 0 . 8

Proof outline Idea: take an optimal partition in R n ( n huge) and try to “simulate” it in R n 0 . 1. An optimal partition is close to a bounded-degree polynomial threshold function (PTF) 8

Proof outline Idea: take an optimal partition in R n ( n huge) and try to “simulate” it in R n 0 . 1. An optimal partition is close to a bounded-degree polynomial threshold function (PTF) 2. A bounded-degree PTF can be approximately simulated by a bounded-degree PTF on a bounded number of variables 8

Step 1: approximation by polynomials

Approximation by polynomials Think of a partition as a function f : R n → { e 1 , . . . , e k } ⊂ R k . 9

Approximation by polynomials Think of a partition as a function f : R n → { e 1 , . . . , e k } ⊂ R k . ˆ � Hermite expansion f ( x ) = f α, i H α ( x ) e i α, i 9

Approximation by polynomials Think of a partition as a function f : R n → { e 1 , . . . , e k } ⊂ R k . ˆ � Hermite expansion f ( x ) = f α, i H α ( x ) e i α, i ρ deg( H α ) ˆ ˆ � f 2 � f 2 Facts: 1 = α, i and Stab ρ ( f ) = α, i α, i α, i 9

Approximation by polynomials Think of a partition as a function f : R n → { e 1 , . . . , e k } ⊂ R k . ˆ � Hermite expansion f ( x ) = f α, i H α ( x ) e i α, i ρ deg( H α ) ˆ ˆ � f 2 � f 2 Facts: 1 = α, i and Stab ρ ( f ) = α, i α, i α, i Idea: noise stability ⇒ lots of “low-degree” weight ⇒ approximate f by truncating the expansion 9

Approximation by polynomials Think of a partition as a function f : R n → { e 1 , . . . , e k } ⊂ R k . ˆ � Hermite expansion f ( x ) = f α, i H α ( x ) e i α, i ρ deg( H α ) ˆ ˆ � f 2 � f 2 Facts: 1 = α, i and Stab ρ ( f ) = α, i α, i α, i Idea: noise stability ⇒ lots of “low-degree” weight ⇒ approximate f by truncating the expansion Real proof goes through a smoothing/rounding procedure, and a connection with Gaussian surface area (KNOW ’15). 9

Step 2: dimension reduction

Polynomial structure theorem (De-Servedio) where ℓ is bounded and v 1 , . . . , v ℓ are “nice” 10

Polynomial Central Limit Theorem (De-Servedio) 11

Polynomial Central Limit Theorem (De-Servedio) “Nice” polynomials satisfy a CLT, so they may as well be linear functions of ℓ variables 11

Polynomial Central Limit Theorem (De-Servedio) “Nice” polynomials satisfy a CLT, so they may as well be linear functions of ℓ variables 11

Open problem Conjecture (Peace sign conjecture) The optimal partition looks like a peace sign. 12

Thank you!