Abstract
In this paper we studynoninteractive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model toNICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating.
Our new contributions include the following:

• In the case of akleaf star graph (the model considered in [31]), we resolve the open question of whether the success probability must go to zero ask » ∞. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowlydecaying polynomial).

• In the case of thekvertex path graph, we show that it is always optimal for all players to use the same 1bit function.

• In the general case we show that all players should use monotone functions. We also show, somewhat surprisingly, that for certain trees it is better if not all players use the same function.
Our techniques include the use of thereverse BonamiBeckner inequality. Although the usual BonamiBeckner has been frequently used before, its reverse counterpart seems not to be well known. To demonstrate its strength, we use it to prove a new isoperimetric inequality for the discrete cube and a new result on the mixing of short random walks on the cube. Another tool that we need is a tight bound on the probability that a Markov chain stays inside certain sets; we prove a new theorem generalizing and strengthening previous such bounds [2, 3, 6]. On the probabilistic side, we use the “reflection principle” and the FKG and related inequalities in order to study the problem on general trees.
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References
M. Abramowitz and I. Stegun,Handbook of Mathematical Functions, Dover, New York, 1972.
M. Ajtai, J. Komolós and E. Szemerédi,Deterministic simulation in LOGSPACE, inProceedings of the 19th Annual ACM Symposium on Theory of Computing, ACM, New York, 1987, pp. 132–140.
N. Alon, U. Feige, A. Wigderson and D. Zuckerman,Derandomized graph products, Computational Complexity, Birkhäuser, Basel, 1995.
N. Alon, G. Kalai, M. Ricklin and L. Stockmeyer,Lower bounds on the competitive ratio for mobile user tracking and distributed job scheduling, Theoretical Computer Science130 (1994), 175–201.
N. Alon, U. Maurer and A. Wigderson, Unpublished results, 1991.
N. Alon and J. Spencer,The Probabilistic Method, 2nd edn., Wiley, New York, 2000.
K. Amano and A. Maruoka,On learning monotone Boolean functions under the uniform distribution, Lecture Notes in Computer Science2533, Springer, New York, 2002, pp. 57–68.
W. Beckner,Ineqalities in Fourier analysis, Annals of Mathematics102 (1975), 159–182.
M. BenOr and N. Linial,Collective coin flipping, inRandomness and Computation (S. Micali, ed.), Academic Press, New York, 1990.
I. Benjamini, G. Kalai and O Schramm,Noise sensitivity of boolean functions and applications to percolation, Publications Mathématiques de l'Institut des Hautes Études Scientifiques90 (1999), 5–43.
S. Bobkov and F. Götze,Discrete isoperimetric, and Poincarétype inequalities, Probability theory and Related Fields114 (1999), 245–277.
A. Bonami,Études des coefficients Fourier des fonctiones de L ^{p} (G), Annales de l'Institut Fourier20 (1970), 335–402.
C. Borell,Positivity improving operators and hypercontractivity, Mathematische Zeitschrift180 (1982), 225–234.
J. Bourgain, An appendix toSharp thresholds of graph properties, and the ksat problem, by E. Friedgut, Journal of the American Mathematical Society12 (1999), 1017–1054.
J. Bourgain, J. Kahn, G. Kalai, Y. Katznelson and N. Linial,The influence of variables in product spaces. Israel Journal of Mathematics77 (1992), 55–64.
J. Bourgain and G. Kalai,Influences of variables and threshold intervals under group symmetries, Geometric and Functional Analysis7 (1997), 438–461.
N. Bshouty, J. Jackson and C. Tamon,Uniformdistribution attribute noise learnability, inProceedings of the Eighth Annual Conference on Computational Learning Theory, (COLT 1995), Santa Cruz, California, USA, ACM, 1995.
I. Dinur, V. Guruswami and S. Khot,Vertex cover on kuniform hypergraphs is hard to approximate within factor (k3∈), ECCC Technical Report TRO2027, 2002.
I. Dinur and S. Safra,The importance of being biased, inProceedings of the 34th Annual ACM Symposium on the Theory of Computing, ACM, New York, 2002, pp. 33–42.
W. Feller,An Introduction to Probability Theory and its Applications, 3rd edn., Wiley, New York, 1968.
C. Fortuin, P. Kasteleyn and J. Ginibre,Correlation inequalities on some partially ordered sets, Communications in Mathematical Physics22 (1971), 89–103.
E. Friedgut,Boolean functions with low average sensivity depend on few coordinates, Combinatorica18 (1998), 474–483.
E. Friedgut and G. Kalai,Every monotone graph property has a sharp threshold, Proceedings of the American Mathematical Society124 (1996), 2993–3002.
H. O. Georgii,Gibbs Measures and Phase Transitions, Volume 9 of de Gruyter Studies in Mathematics, de Gruyter, Berlin, 1988.
G. Hardy, J. Littlewood and G. Póyla,Inequalities, 2nd edn., Cambridge University Press, 1952.
J. H»stad,Some optimal inapproximability results, Journal of the ACM48 (2001), 798–869.
J. Kahn, G. Kalai and N. LinialThe influence of variables on boolean functions, inProceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1988, pp. 68–80.
S. Khot,On the power of unique 2prover 1round games, inProceedings of the 34th Annual ACM Symposium on the Theory of Computing, ACM, New York, 2002, pp. 767–775.
D. Kleitman,Families of nondisjoint subsets, Journal of Combinatorial Theory1 (1966), 153–155.
A. Klivans, R. O'Donnell and R. Servedio,Learning intersections and thresholds of halfspaces, inProceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 2002, pp. 177–186.
E. Mossel and R. O'Donnell,Coin flipping from a cosmic source: On error correction of truly random bits, Random Structures & Algorithms26 (2005), 418–436.
A. Naor, E. Friedgut and G. Kalai,Boolean functions whose Fourier transform is concentrated on the first two levels, Advances in Applied Mathematics29 (2002), 427–437.
R. O'Donnell,Hardness amplification within NP, inProceedings of the 34th Annual ACM Symposium on the Theory of Computing, ACM, New York, 2002, pp. 715–760.
R. O'Donnell,Computational applications of noise sensitivity, PhD thesis, Massachusetts Institute of Technology, 2003.
R. O'Donnell and R. Servedio,Learning monotone decision trees, Manuscript, 2004.
R. Raz,Fourier analysis for probabilistic communication complexity, Computational Complexity5 (1995), 205–221.
V. Sazonov,Normal Approximation— Some Recent Advances, SpringerVerlag, Berlin, 1981.
M. Talagrand,On Russo's approximate 0–1 law, Annals of Probability22 (1994), 1476–1387.
K. Yang,On the (im)possibility of noninteractive correlation distillation, inLATIN 2004: Theoretical Informatics, 6th Latin American Symposium, Buenos Aires, Argentina, April 5–8, 2004, Proceedings (M. FarachColton, ed.), Lecture Notes in Computer Science 2976, Springer, Berlin, 2004.
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Supported by a Miller fellowship in Statistics and CS, U.C. Berkeley, by an Alfred P. Sloan fellowship in Mathematics, and by NSF grant DMS0504245.
Most of this work was done while the author was a student at Massachusetts Institute of Technology. This material is based upon work supported by the National Science Foundation under agreement No. CCR0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Most of this work was done while the author was at the Institute for Advanced Study, Princeton, NJ. Work supported by an Alon Fellowship, ARO grant DAAD190310082 and NSF grant CCR9987845.
Supported in part by NSF grant DMS0103841 the Swedish Research Council and the Göran Gustafsson Foundation (KVA).
Research supported in part by NSF grant DMS0106589, DMS0355497, and by an Alfred P. Sloan fellowship.
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Mossel, E., O'Donnell, R., Regev, O. et al. Noninteractive correlation distillation, inhomogeneous Markov chains, and the reverse BonamiBeckner inequality. Isr. J. Math. 154, 299–336 (2006). https://doi.org/10.1007/BF02773611
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DOI: https://doi.org/10.1007/BF02773611