
Optimal Selection for Good Polynomials of Degree up to Five
Good polynomials are the fundamental objects in the TamoBarg constructi...
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Distributed Source Simulation With No Communication
We consider the problem of distributed source simulation with no communi...
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Bounds for the asymptotic distribution of the likelihood ratio
In this paper we give an explicit bound on the distance to chisquare for...
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Boolean functions: noise stability, noninteractive correlation, and mutual information
Let ϵ∈[0, 1/2] be the noise parameter and p>1. We study the isoperimetri...
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On certain linearized polynomials with high degree and kernel of small dimension
Let f be the F_qlinear map over F_q^2n defined by x x+ax^q^s+bx^q^n+s w...
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Three Candidate Plurality is Stablest for Small Correlations
Using the calculus of variations, we prove the following structure theor...
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Dimension Reduction for Polynomials over Gaussian Space and Applications
We introduce a new technique for reducing the dimension of the ambient space of lowdegree polynomials in the Gaussian space while preserving their relative correlation structure, analogous to the JohnsonLindenstrauss lemma. As applications, we address the following problems: 1. Computability of Approximately Optimal Noise Stable function over Gaussian space: The goal is to find a partition of R^n into k parts, that maximizes the noise stability. An δoptimal partition is one which is within additive δ of the optimal noise stability. De, Mossel & Neeman (CCC 2017) raised the question of proving a computable bound on the dimension n_0(δ) in which we can find an δoptimal partition. While De et al. provide such a bound, using our new technique, we obtain improved explicit bounds on the dimension n_0(δ). 2. Decidability of NonInteractive Simulation of Joint Distributions: A "noninteractive simulation" problem is specified by two distributions P(x,y) and Q(u,v): The goal is to determine if two players that observe sequences X^n and Y^n respectively where {(X_i, Y_i)}_i=1^n are drawn i.i.d. from P(x,y) can generate pairs U and V respectively (without communicating with each other) with a joint distribution that is arbitrarily close in total variation to Q(u,v). Even when P and Q are extremely simple, it is open in several cases if P can simulate Q. In the special where Q is a joint distribution over {0,1}×{0,1}, Ghazi, Kamath and Sudan (FOCS 2016) proved a computable bound on the number of samples n_0(δ) that can be drawn from P(x,y) to get δclose to Q (if it is possible at all). Recently De, Mossel & Neeman obtained such bounds when Q is a distribution over [k] × [k] for any k > 2. We recover this result with improved explicit bounds on n_0(δ).
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