A Variational Proof of Robust Gaussian Noise Stability

Heilman, Steven

doi:10.48550/arxiv.2108.04950

Cited by 2 publications

(3 citation statements)

References 17 publications

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“…specifying which sets maximize (8) in the case m = 2. Later, we showed in [Hei21c] that this strategy could be extended to prove a robust version of Borell's inequality, proving some conjectures of Eldan [Eld15]. In that result, we consider noise stability plus a "penalty term," and we show that if this penalty term is sufficiently small, then the sets maximizing noise stability still maximize the noise stability plus a penalty term.…”

Section: Introductionmentioning

confidence: 71%

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Dimension-Free Noninteractive Simulation from Gaussian Sources

Heilman¹,

Tarter²

2022

Preprint

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Let X and Y be two real-valued random variables. Let (X 1 , Y 1 ), (X 2 , Y 2 ), . . . be independent identically distributed copies of (X, Y ). Suppose there are two players A and B. Player A has access to X 1 , X 2 , . . . and player B has access to Y 1 , Y 2 , . . .. Without communication, what joint probability distributions can players A and B jointly simulate? That is, if k, m are fixed positive integers, what probability distributions on {1, . . . , m} 2 are equal to the distribution ofWhen X and Y are standard Gaussians with fixed correlation ρ ∈ (−1, 1), we show that the set of probability distributions that can be noninteractively simulated from k Gaussian samples is the same for any k ≥ m 2 . Previously, it was not even known if this number of samples m 2 would be finite or not, except when m ≤ 2.Consequently, a straightforward brute-force search deciding whether or not a probability distribution on {1, . . . , m} 2 is within distance 0 < ε < |ρ| of being noninteractively simulated from k correlated Gaussian samples has run time bounded by (5/ε) m(log(ε/2)/ log |ρ|) m 2 , improving a bound of Ghazi, Kamath and Raghavendra.A nonlinear central limit theorem (i.e. invariance principle) of Mossel then generalizes this result to decide whether or not a probability distribution on {1, . . . , m} 2 is within distance 0 < ε < |ρ| of being noninteractively simulated from k samples of a given finite discrete distribution (X, Y ) in run time that does not depend on k, with constants that again improve a bound of Ghazi, Kamath and Raghavendra.

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Section: Introductionmentioning

confidence: 71%

“…The strategy of [HT21,Hei21c] was recently adapted in [HNP + 21] to the setting of functions taking values in spheres, rather than functions taking values in the simplex.…”

Section: Introductionmentioning

confidence: 99%

Dimension-Free Noninteractive Simulation from Gaussian Sources

Heilman¹,

Tarter²

2022

Preprint

Self Cite

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“…It was unclear if the methods of [CM12] could apply to the noise stability functional in Definition 1.13 until [HT21], where these methods were adapted to show that the k-set Standard Simplex Conjecture in R n reduces to the same conjecture in R k−1 , for any n ≥ k − 1. This proof method can also prove Borell's original inequality [HT21,Hei21]. The result of [HT21] also circumvented a difficulty identified in [HMN16].…”

Section: Introductionmentioning

confidence: 86%

Three candidate plurality is stablest for small correlations

Heilman

Tarter

2021

Forum of Mathematics, Sigma

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Using the calculus of variations, we prove the following structure theorem for noise-stable partitions: a partition of n-dimensional Euclidean space into m disjoint sets of fixed Gaussian volumes that maximise their noise stability must be $(m-1)$ -dimensional, if $m-1\leq n$ . In particular, the maximum noise stability of a partition of m sets in $\mathbb {R}^{n}$ of fixed Gaussian volumes is constant for all n satisfying $n\geq m-1$ . From this result, we obtain: (i) A proof of the plurality is stablest conjecture for three candidate elections, for all correlation parameters $\rho $ satisfying $0<\rho <\rho _{0}$ , where $\rho _{0}>0$ is a fixed constant (that does not depend on the dimension n), when each candidate has an equal chance of winning. (ii) A variational proof of Borell’s inequality (corresponding to the case $m=2$ ). The structure theorem answers a question of De–Mossel–Neeman and of Ghazi–Kamath–Raghavendra. Item (i) is the first proof of any case of the plurality is stablest conjecture of Khot-Kindler-Mossel-O’Donnell for fixed $\rho $ , with the case $\rho \to L1^{-}$ being solved recently. Item (i) is also the first evidence for the optimality of the Frieze–Jerrum semidefinite program for solving MAX-3-CUT, assuming the unique games conjecture. Without the assumption that each candidate has an equal chance of winning in (i), the plurality is stablest conjecture is known to be false.

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A Variational Proof of Robust Gaussian Noise Stability

Cited by 2 publications

References 17 publications

Dimension-Free Noninteractive Simulation from Gaussian Sources

Dimension-Free Noninteractive Simulation from Gaussian Sources

Three candidate plurality is stablest for small correlations

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