Gaussian Bounds for Noise Correlation of Functions

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 multi-dimensional Gaussian distributions.

[1]  On the Number of Catenaries That May be Drawn Through Two Fixed Points , 1895 .

[2]  W. Sheppard.  On the Application of the Theory of Error to Cases of Normal Distribution and Normal Correlation , 1899 .

[3]  J. Lindeberg.  Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung , 1922 .

[4]  H. Gebelein.  Das statistische Problem der Korrelation als Variations‐ und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung , 1941 .

[5]  A. Rényi.  On measures of dependence , 1959 .

[6]  R. Niemi,et al.  A mathematical solution for the probability of the paradox of voting. , 1968, Behavioral science.

[7]  A. Bonami.  Étude des coefficients de Fourier des fonctions de $L^p(G)$ , 1970 .

[8]  W. Beckner.  Inequalities in Fourier analysis , 1975 .

[9]  V. Rotaŕ.  Limit Theorems for Multilinear Forms and Quasipolynomial Functions , 1976 .

[10]  Colin E Bell,et al.  A Random Voting Graph Almost Surely Has a Hamiltonian Cycle When the Number of Alternatives Is Large , 1981 .

[11]  B. Efron,et al.  The Jackknife Estimate of Variance , 1981 .

[12]  C. Borell.  Geometric bounds on the Ornstein-Uhlenbeck velocity process , 1985 .

[13]  J. Szulga,et al.  Hypercontraction principle and random multilinear forms , 1988 .

[14]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[15]  S. Janson.  Gaussian Hilbert Spaces , 1997 .

[16]  D. Nualart.  GAUSSIAN HILBERT SPACES (Cambridge Tracts in Mathematics 129) By SVANTE JANSON: 340 pp., £40.00, ISBN 0 521 56128 0 (Cambridge University Press, 1997) , 1998 .

[17]  Ehud Friedgut,et al.  Boolean Functions With Low Average Sensitivity Depend On Few Coordinates , 1998, Comb..

[18]  Alan M. Frieze,et al.  Quick Approximation to Matrices and Applications , 1999, Comb..

[19]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[20]  W. T. Gowers,et al.  A NEW PROOF OF SZEMER ´ EDI'S THEOREM , 2001 .

[21]  Y. Rinott,et al.  A remark on quadrant normal probabilities in high dimensions , 2001 .

[22]  Irit Dinur,et al.  The importance of being biased , 2002, STOC '02.

[23]  Gil Kalai,et al.  A Fourier-theoretic perspective on the Condorcet paradox and Arrow's theorem , 2002, Adv. Appl. Math..

[24]  Subhash Khot.  On the power of unique 2-prover 1-round games , 2002, STOC '02.

[25]  K. Oleszkiewicz.  On a Nonsymmetric Version of the Khinchine-Kahane Inequality , 2003 .

[26]  B. Green.  A Szemerédi-type regularity lemma in abelian groups, with applications , 2003, math/0310476.

[27]  Guy Kindler,et al.  Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[28]  Gil Kalai,et al.  NOTES AND COMMENTS: SOCIAL INDETERMINACY , 2004 .

[29]  Ryan O'Donnell,et al.  Hardness amplification within NP , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[30]  Ryan O'Donnell,et al.  Noise stability of functions with low influences: Invariance and optimality , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[31]  Venkatesan Guruswami,et al.  A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover , 2005, SIAM J. Comput..

[32]  Terence Tao.  Szemerédi's regularity lemma revisited , 2006, Contributions Discret. Math..

[33]  Luca Trevisan,et al.  Gowers uniformity, influence of variables, and PCPs , 2005, STOC '06.

[34]  P. Wolff,et al.  Hypercontractivity of simple random variables , 2007 .

[35]  Ryan O'Donnell,et al.  Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs? , 2007, SIAM J. Comput..

[36]  Elchanan Mossel,et al.  Noise stability of functions with low influences: Invariance and optimality , 2005, IEEE Annual Symposium on Foundations of Computer Science.

[37]  Prasad Raghavendra,et al.  Optimal algorithms and inapproximability results for every CSP? , 2008, STOC.

[38]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[39]  Elchanan Mossel,et al.  Approximation Resistant Predicates from Pairwise Independence , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[40]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM , 2008, Proceedings of the Edinburgh Mathematical Society.

[41]  Elchanan Mossel,et al.  Conditional Hardness for Approximate Coloring , 2009, SIAM J. Comput..