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Noise Stability is Low-Dimensional Anindya De, Elchanan Mossel, Joe - - PowerPoint PPT Presentation
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 -correlated Gaussians in R n (0 < < 1). For a partition A = ( A 1 , . . . , A k ) of R n into k parts, define
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Gaussian noise stability
Noise stable Not noise stable
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Gaussian noise stability
Theorem (Borell ’85) For a partition of Rn into two parts of equal Gaussian measure, Stabρ(A) ≤ 1 2 + sin−1 ρ π . Equality is attained for a partition into half-spaces.
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Gaussian noise stability
Theorem (Borell ’85) For a partition of Rn into two parts of equal Gaussian measure, Stabρ(A) ≤ 1 2 + sin−1 ρ π . Equality is attained for a partition into half-spaces.
- well-known links to computational complexity (KKMO ’05)
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Gaussian noise stability
Theorem (Borell ’85) For a partition of Rn into two parts of equal Gaussian measure, Stabρ(A) ≤ 1 2 + sin−1 ρ π . Equality is attained for a partition into half-spaces.
- well-known links to computational complexity (KKMO ’05)
- one-dimensional phenomenon
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Gaussian noise stability
Theorem (???) For a partition of Rn into three parts of equal Gaussian measure, ???
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Gaussian noise stability
Conjecture (Peace sign conjecture) The optimal partition looks like a peace sign.
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Gaussian noise stability
Conjecture (Peace sign conjecture) The optimal partition looks like a peace sign. (two-dimensional phenomenon)
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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 Rk−1. It looks like a multi-dimensional peace sign.
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Gaussian noise stability
Theorem (De-Mossel-N.) For any k and any ǫ > 0, there is a computable n0 = n0(k, ǫ) such that an ǫ-approximately optimal partition occurs in Rn0.
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Gaussian noise stability
Theorem (De-Mossel-N.) For any k and any ǫ > 0, there is a computable n0 = n0(k, ǫ) such that an ǫ-approximately optimal partition occurs in Rn0. Corollary The optimal value of k-part noise stability is computable.
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Gaussian noise stability
Theorem (De-Mossel-N.) For any k and any ǫ > 0, there is a computable n0 = n0(k, ǫ) such that an ǫ-approximately optimal partition occurs in Rn0. 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.
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Correlation distillation
Goal: produce uniform output, agree with maximal probability. What is the probability of agreement?
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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
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The main theorem
Theorem (De-Mossel-N.) For any k and any ǫ > 0, there is a computable n0 = n0(k, ǫ) such that an ǫ-approximately optimal partition occurs in Rn0.
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Proof outline
Idea: take an optimal partition in Rn (n huge) and try to “simulate” it in Rn0.
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Proof outline
Idea: take an optimal partition in Rn (n huge) and try to “simulate” it in Rn0.
- 1. An optimal partition is close to a bounded-degree polynomial
threshold function (PTF)
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Proof outline
Idea: take an optimal partition in Rn (n huge) and try to “simulate” it in Rn0.
- 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
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Step 1: approximation by polynomials
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Approximation by polynomials
Think of a partition as a function f : Rn → {e1, . . . , ek} ⊂ Rk.
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Approximation by polynomials
Think of a partition as a function f : Rn → {e1, . . . , ek} ⊂ Rk. Hermite expansion f (x) =
- α,i
ˆ fα,iHα(x)ei
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Approximation by polynomials
Think of a partition as a function f : Rn → {e1, . . . , ek} ⊂ Rk. Hermite expansion f (x) =
- α,i
ˆ fα,iHα(x)ei Facts: 1 =
- α,i
ˆ f 2
α,i and Stabρ(f ) =
- α,i
ρdeg(Hα) ˆ f 2
α,i 9
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Approximation by polynomials
Think of a partition as a function f : Rn → {e1, . . . , ek} ⊂ Rk. Hermite expansion f (x) =
- α,i
ˆ fα,iHα(x)ei Facts: 1 =
- α,i
ˆ f 2
α,i and Stabρ(f ) =
- α,i
ρdeg(Hα) ˆ f 2
α,i
Idea: noise stability ⇒ lots of “low-degree” weight ⇒ approximate f by truncating the expansion
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Approximation by polynomials
Think of a partition as a function f : Rn → {e1, . . . , ek} ⊂ Rk. Hermite expansion f (x) =
- α,i
ˆ fα,iHα(x)ei Facts: 1 =
- α,i
ˆ f 2
α,i and Stabρ(f ) =
- α,i
ρdeg(Hα) ˆ f 2
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).
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Step 2: dimension reduction
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Polynomial structure theorem (De-Servedio)
where ℓ is bounded and v1, . . . , vℓ are “nice”
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Polynomial Central Limit Theorem (De-Servedio)
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Polynomial Central Limit Theorem (De-Servedio)
“Nice” polynomials satisfy a CLT, so they may as well be linear functions of ℓ variables
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Polynomial Central Limit Theorem (De-Servedio)
“Nice” polynomials satisfy a CLT, so they may as well be linear functions of ℓ variables
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Open problem
Conjecture (Peace sign conjecture) The optimal partition looks like a peace sign.
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