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Gaussian Bounds for Noise Correlation of Functions and Tight Analysis of Long Codes
 In IEEE Symposium on Foundations of Computer Science (FOCS
, 2008
"... In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recentl ..."
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Cited by 35 (5 self)
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In this paper we derive tight bounds on the expected value of products of low influence functions defined on correlated probability spaces. The proofs are based on extending Fourier theory to an arbitrary number of correlated probability spaces, on a generalization of an invariance principle recently obtained with O’Donnell and Oleszkiewicz for multilinear polynomials with low influences and bounded degree and on properties of multidimensional Gaussian distributions. We present two applications of the new bounds to the theory of social choice. We show that Majority is asymptotically the most predictable function among all low influence functions given a random sample of the voters. Moreover, we derive an almost tight bound in the context of Condorcet aggregation and low influence voting schemes on a large number of candidates. In particular, we show that for every low influence aggregation function, the probability that Condorcet voting on k candidates will result in a unique candidate that is preferable to all others is k−1+o(1). This matches the asymptotic behavior of the majority function for which the probability is k−1−o(1). A number of applications in hardness of approximation in theoretical computer science were
Contraction And Decoupling Inequalities For Multilinear Forms And UStatistics
, 1994
"... this paper, we prove decoupling inequalities for random variables that are not necessarily symmetric. Theorems 2.1 and 2.3 in Section 2, and Theorem 3.8 in Section 3, are our main results. The decoupling principle by means of probability tails, Theorem 3.8, immediately ensures the parity of tightnes ..."
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Cited by 9 (3 self)
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this paper, we prove decoupling inequalities for random variables that are not necessarily symmetric. Theorems 2.1 and 2.3 in Section 2, and Theorem 3.8 in Section 3, are our main results. The decoupling principle by means of probability tails, Theorem 3.8, immediately ensures the parity of tightness of two types of chaoses (that Gaussian decoupled and coupled chaoses are simultaneously tight was proved in [Kwa87]). A number of decoupling results are obtained for arbitrary rearrangement invariant norms and Orlicz functionals. In particular, we provide one extended example regarding certain Lorentz norms (important in the approximation theory). Another application is the decoupling principle for Ustatistics (a result as in Theorem 2.3 was proven in [dlPn92]). The utilized techniques are based on ideas, borrowed from [Kwa87], while some are taken from [KW92]. Proofs are straightforward and point out the algebraic nature of decoupling that is fruitfully merged with a widely understood context of convexity. A rule of thumb is that, in the eld of random diagonalfree polynomials, a \denable" is \decouplable". The obtained robust constants are tightly estimated, and are sharper than constants known before. In the last section, we show tail probability decoupling results for polynomials of symmetric random variables. This section makes use of techniques from [AMS92]. CONTRACTION AND DECOUPLING 3 1.2 Notation Random variables in this paper are dened on a separable probability space(
Fooling Functions of Halfspaces under Product Distributions
, 2010
"... We construct pseudorandom generators that fool functions of halfspaces (threshold functions) under a very broad class of product distributions. This class includes not only familiar cases such as the uniform distribution on the discrete cube, the uniform distribution on the solid cube, and the multi ..."
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Cited by 7 (1 self)
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We construct pseudorandom generators that fool functions of halfspaces (threshold functions) under a very broad class of product distributions. This class includes not only familiar cases such as the uniform distribution on the discrete cube, the uniform distribution on the solid cube, and the multivariate Gaussian distribution, but also includes any product of discrete distributions with probabilities bounded away from 0. Our first main result shows that a recent pseudorandom generator construction of Meka and Zuckerman [MZ09], when suitably modified, can fool arbitrary functions of d halfspaces under product distributions where each coordinate has bounded fourth moment. To ɛfool any sizes, depthd decision tree of halfspaces, our pseudorandom generator uses seed length O((d log(ds/ɛ) + log n) ∙ log(ds/ɛ)). For monotone functions of d halfspaces, the seed length can be improved to O((d log(d/ɛ) + log n) ∙ log(d/ɛ)). We get better bounds for larger ɛ; for example, to 1/polylog(n)fool all monotone functions of (log n) / log log n halfspaces, our generator requires a seed of length just O(log n).
Hypercontractivity and Comparison of Moments of Iterated Maxima and Minima of Independent Random Variables
, 1998
"... : We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball p ..."
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Cited by 6 (3 self)
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: We provide necessary and sufficient conditions for hypercontractivity of the minima of nonnegative, i.i.d. random variables and of both the maxima of minima and the minima of maxima for such r.v.'s. It turns out that the idea of hypercontractivity for minima is closely related to small ball probabilities and Gaussian correlation inequalities. Keywords: hypercontractivity, comparison of moments, iterated maxima and minima, Gaussian correlation inequalities, small ball probabilities. Submitted to EJP on May 17, 1997. Final version accepted on January 7, 1998. AMS 1991 Subject Classification: Primary 60B11, 60E07, 60E15. Secondary 52A21, 60G15 * Supported in part by an NSF grant y Participant, NSF Workshop in Linear Analysis & Probability, Texas A&M University ** Supported in part by Polish KBN grant ? Supported in part by USIsrael Binational Science Foundation z Supported in part by a Texas Advanced Research Program grant 1 Section 1. Introduction. Let h be a real valued fu...
Gaussian Noise Sensitivity and Fourier Tails
, 2011
"... We observe a subadditivity property for the noise sensitivity of subsets of Gaussian space. For subsets of volume 1 2, this leads to an almost trivial proof of Borell’s Isoperimetric Inequality for ρ = cos ( π), ℓ ∈ N. In turn this can be used to obtain the Gaussian Isoperimetric Inequality for ..."
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Cited by 3 (1 self)
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We observe a subadditivity property for the noise sensitivity of subsets of Gaussian space. For subsets of volume 1 2, this leads to an almost trivial proof of Borell’s Isoperimetric Inequality for ρ = cos ( π), ℓ ∈ N. In turn this can be used to obtain the Gaussian Isoperimetric Inequality for
Hypercontractivity and comparison of moments of iterated maxima and minima of independent . . .
, 1998
"... ..."
1 Infinite order decoupling of random chaoses in Banach space
, 1992
"... We prove a number of decoupling inequalities for nonhomogeneous random polynomials with coefficients in Banach space. Degrees of homogeneous components enter into comparison as exponents of multipliers of terms of certain Poincarétype polynomials. It turns out that the fulfillment of most of types ..."
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We prove a number of decoupling inequalities for nonhomogeneous random polynomials with coefficients in Banach space. Degrees of homogeneous components enter into comparison as exponents of multipliers of terms of certain Poincarétype polynomials. It turns out that the fulfillment of most of types of decoupling inequalities may depend on the
Hypercontractive inequalities via SOS, and the Frankl–Rödl graph
, 2013
"... Our main result is a formulation and proof of the reverse hypercontractive inequality in the sumofsquares (SOS) proof system. As a consequence we show that for any constant 0 < γ ≤ 1/4, the ⌉ certifies the statement “the maximum independent set in SOS/Lasserre SDP hierarchy at degree 4 ⌈ 1 4γ th ..."
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Our main result is a formulation and proof of the reverse hypercontractive inequality in the sumofsquares (SOS) proof system. As a consequence we show that for any constant 0 < γ ≤ 1/4, the ⌉ certifies the statement “the maximum independent set in SOS/Lasserre SDP hierarchy at degree 4 ⌈ 1 4γ the Frankl–Rödl graph FR n γ has fractional size o(1)”. Here FR n γ = (V, E) is the graph with V = {0, 1} n and (x, y) ∈ E whenever ∆(x, y) = (1 − γ)n (an even integer). In particular, we show the degree4 SOS algorithm certifies the chromatic number lower bound “χ(FR n 1/4) = ω(1)”, even though FR n 1/4 is the canonical integrality gap instance for which standard SDP relaxations cannot even certify “χ(FR n 1/4)> 3”. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)hypercontractive inequality for any even integer q. 1
1 CONTRACTION AND DECOUPLING INEQUALITIES FOR MULTILINEAR FORMS AND USTATISTICS
, 1999
"... We prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space. We use various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity, etc. ..."
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We prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space. We use various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity, etc.