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11
Optimal inapproximability results for MAXCUT and other 2variable CSPs?
, 2005
"... In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games ..."
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Cited by 173 (24 self)
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In this paper we show a reduction from the Unique Games problem to the problem of approximating MAXCUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the GoemansWilliamson algorithm [25]. This implies that if the Unique Games
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 37 (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
Some Topics in Analysis of Boolean Functions
"... This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow’s Theorem and other ideas from the theory of Social Choice; the BonamiBeckner Inequality as an exten ..."
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Cited by 29 (0 self)
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This article accompanies a tutorial talk given at the 40th ACM STOC conference. In it, we give a brief introduction to Fourier analysis of boolean functions and then discuss some applications: Arrow’s Theorem and other ideas from the theory of Social Choice; the BonamiBeckner Inequality as an extension of Chernoff/Hoeffding bounds to higherdegree polynomials; and, hardness for approximation algorithms.
Noise sensitivity and chaos in social choice theory, preprint
, 2005
"... In this paper we study the social preferences obtained from monotone neutral social welfare functions for random individual preferences. We identify a class of social welfare functions that demonstrate a completely chaotic behavior: they lead to a uniform probability distribution on all possible soc ..."
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Cited by 8 (4 self)
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In this paper we study the social preferences obtained from monotone neutral social welfare functions for random individual preferences. We identify a class of social welfare functions that demonstrate a completely chaotic behavior: they lead to a uniform probability distribution on all possible social preference relations and, for every ɛ> 0, if a small fraction ɛ of individuals change their preferences (randomly) the correlation between the resulting social preferences and the original ones tends to zero as the number of individuals in the society increases. This class includes natural multilevel majority rules.
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 (0 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
18.177 course project: Invariance Principles
"... An invariance principle is a result permitting us to change our underlying probability space—such as occurs in a central limit theorem. For example, we can suggestively state the BerryEssen Theorem in the following way: Theorem 1.1 (BerryEssen) Let X1,..., Xn be i.i.d. random variables with E[Xi] ..."
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An invariance principle is a result permitting us to change our underlying probability space—such as occurs in a central limit theorem. For example, we can suggestively state the BerryEssen Theorem in the following way: Theorem 1.1 (BerryEssen) Let X1,..., Xn be i.i.d. random variables with E[Xi] = 0, E[X2 i] =
E. E. Slutsky Theory of Correlation and Elements of the Doctrine of the Curves of Distribution Manual for Studying Some Most Important Methods of Contemporary Statistics Translated by Oscar Sheynin
"... Теория корреляции и элементы учения о кривых распределения ..."
NOISE STABILITY OF WEIGHTED MAJORITY
, 2004
"... Abstract. Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of n independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability ǫ, the probability pǫ that the weighted majority changes is at most Cǫ 1/4. They asked what is the best pos ..."
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Abstract. Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of n independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability ǫ, the probability pǫ that the weighted majority changes is at most Cǫ 1/4. They asked what is the best possible exponent that could replace 1/4. We prove that the answer is 1/2. The upper bound obtained for pǫ is within a factor of √ π/2 + o(1) from the known lower bound when ǫ → 0 and nǫ → ∞. 1.
max wC Pr [L(S) satisfies R], L∼λC
"... subject to the following conditions: • for all C, λC is a probability distribution on assignments S → D; • (Iv[ℓ])v∈V,ℓ∈D are joint random variables (which can be thought of as vectors), called “pseudoindicators, ” that satisfy: 1. [optional] for all C ∈ C, for all v ∈ C, and for all ℓ ∈ D, E [ Iv[ℓ ..."
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subject to the following conditions: • for all C, λC is a probability distribution on assignments S → D; • (Iv[ℓ])v∈V,ℓ∈D are joint random variables (which can be thought of as vectors), called “pseudoindicators, ” that satisfy: 1. [optional] for all C ∈ C, for all v ∈ C, and for all ℓ ∈ D, E [ Iv[ℓ] ] = Pr [L(v) = ℓ]; L∼λC 2. for all C ∈ C, for all v ∈ C, for all ℓ ∈ D, for all v ′ ∈ V, and for all ℓ ′ ∈ D, E [ Iv[ℓ] · Iv ′[ℓ′] ] = Pr [L(v) = ℓ and L(v L∼λC ′) = ℓ ′]. Also recall the following equivalent perspectives on the canonical SDP relaxation for a CSP: • pseudoindicator random variables satisfying conditions 1 and 2 above; • pseudoindicator random variables satisfying condition 2 above; • a vector solution satisfying conditions 1 and 2 above (when viewed as a collection of pseudoindicator random variables); • jointly Gaussian pseudoindicators satisfying conditions 1 and 2 above.