Abstract
Gaussian bounds on noise correlation of functions play an important role in hardness of approximation, in quantitative social choice theory and in testing. The author (2008) obtained sharp Gaussian bounds for the expected correlation of ℓ low influence functions f(1), …, f(ℓ) : Ωn → [0, 1], where the inputs to the functions are correlated via the n-fold tensor of distribution \({\cal P}\) on Ωℓ in the following way: For each 1 ≤ i ≤ n, the vector consisting of the i’-th inputs to the ℓ functions is sampled according to \({\cal P}\).
It is natural to ask if the condition of low influences can be relaxed to the condition that the function has vanishing Fourier coefficients. Here and g we further show that if f, g have a noisy inner product that exceeds the Gaussian bound, then the Fourier supports of their large coefficients intersect.
Similar content being viewed by others
References
P. Austrin, Towards sharp inapproximability for any 2-CSP, SIAM Journal on Computing 39 (2010), 2430–2463.
P. Austrin and J. Håstad, Randomly supported independence and resistance, SIAM Journal on Computing 40 (2011), 1–27.
P. Austrin and J. Håstad, On the usefulness of predicates, in 2012 IEEE 27th Conference on Computational Complexity—CCC 2012, IEEE Computer Society, Los Alamitos, CA, 2012, pp. 53–63.
P. Austrin and E. Mossel, Noise correlation bounds for uniform low degree functions, Arkiv för Matematik 51 (2013), 29–52.
N. Bansal and S. Khot, Inapproximability ofhypergraph vertex cover and applications to scheduling problems, in Automata, Languages and Programming. Part I, Lecture Notes in Computer Science, Vol. 6198, Springer, Berlin, 2010, pp. 250–261.
E. Biais, Testing juntas nearly optimally, in STOC’09—Proceedings of the 2009 ACM international Symposium on Theory of Computing, ACM, New York, 2009, pp. 151–158.
C. Borell, Geometric bounds on the Ornstein-Uhlenbeck velocity process, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 70 (1985), 1–13.
S. O. Chan, Approximation resistance from pairwise-independent subgroups, Journal of the ACM 63 (2016), Article no. 27.
X. Chen, A. De, R. A. Servedio and L.-Y. Tan, Boolean function monotonicity testing requires (almost) n1/2non-adaptive queries, in STOC’15—Proceedings of the 2015 ACM Symposium on Theory of Computing, ACM, New York, 2015, pp. 519–528.
X. Chen, R. A. Servedio and L.-Y. Tan, New algorithms and lower bounds for monotonicity testing, in 55th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2014, IEEE Computer Society, Los Alamitos, CA, 2014, pp. 286–295.
B. Chor, O. Goldreich, J. Hasted, J. Freidmann, S. Rudich and R. Smolensky, The bit extraction problem or t-resilient functions, in 26th Annual IEEE Symposium on Foundations of Computer Science—FOCS 1985, IEEE Computer Society, Los Alamitos, CA, 1985, pp. 396–407.
I. Diakonikolas, D. M. Kane and J. Nelson, Bounded independence fools degree-2 threshold functions, in 51st Annual IEEE Symposium on Foundations of Computer Science—FOCS 2010, IEEE Computer Society, Los Alamitos, CA, 2010, pp. 11–20. IEEE, 2010.
V. Feldman, V. Guruswami, P. Raghavendra and Y. Wu, Agnostic learning of monomials by halfspaces is hard, SIAM Journal on Computing 41 (2012), 1558–1590.
Y. Filmus, H. Hatami, S. Heilman, E. Mossel, R. O’Donnell, S. Sachdeva, A. Wan and K. Wimmer, Real analysis in computer science: A collection of open problems, https://doi.org/simons.berkeley.edu/sites/default/files/openprobsmerged.pdf.
E. Friedgut, G. Kalai, N. Keller and N. Nisan, A quantitative version of the Gihbard Satterthwaite theorem for three alternatives, SIAM Journal on Computing 40 (2011), 934–952.
V. Guruswami, J. Håstad, R. Manokaran, P. Raghavendra and M. Charikar, Beating the random ordering is hard: Every ordering CSP is approximation resistant, SIAM Journal on Computing 40 (2011), 878–914.
V. Guruswami, R. Manokaran and P. Raghavendra, Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph, in 49th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2008, IEEE Computer Society, Los Alamitos, CA, 2008, pp. 573–582.
V. Guruswami, P. Raghavendra, R. Saket and Y. Wu, Bypassing UGC from some optimal geometric inapproximability results, in Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, pp. 699–717.
J. Håstad, Some optimal inapproximability results, in Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC), ACM, New York, 1999, pp. 1–10.
J. Hązła, T. Holenstein and E. Mossel, Lower bounds on same-set inner product in correlated spaces. in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Leibniz International Proceedings in Informatics, Vol. 34, Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern, 2016, Article no. 34.
M. Isaksson and E. Mossel, New maximally stable Gaussian partitions with discrete applications, Israel Journal of Mathematics 189 (2012), 347–396.
C. Jones, A noisy-inñuence regularity lemma for boolean functions, https://doi.org/abs/1610.06950.
G. Kalai, A Fourier-theoretic perspective on the Condorcet paradox and Arrow’s theorem, Advances in Applied Mathematics 29 (2002), 412–426.
S. Khot, On the power of unique 2-prover 1-round games, in Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 767–775.
S. Khot, G. Kindler, E. Mossel and R. O’Donnell, Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?, in 45th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2004, IEEE Computer Society, Los Alamitos, CA, 2004, pp. 146–154.
S. Khot, G. Kindler, E. Mossel and R. O’Donnell, Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?, S1AM Journal on Computing 37 (2007), 319–357.
E. Mossel, Gaussian bounds for noise correlation of functions and tight analysis of long codes, in 49th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2008, IEEE Computer Society, Los Alamitos, CA, 2008, pp. 156–165.
E. Mossel, Gaussian bounds for noise correlation of functions, Geometric and Functional Analysis 19 (2010), 1713–1756.
E. Mossel, A quantitative arrow theorem, Probability Theory and Related Fields 154 (2012), 49–88.
E. Mossel, R. O’Donnell and K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality (extended abstract), in 46th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2005, IEEE Computer Society, Los Alamitos, 2005, pp. 21–30.
E. Mossel, R. O’Donnell and K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality, Annals of Mathematics 171 (2010), 295–341.
E. Mossel and O. Schramm, Representations of general functions using smooth functions, unpublished manuscript.
I. Nourdin, G. Peccati and G. Reinert, Invariance principies for homogeneous sums: universality of Gaussian Wiener chaos, Annals of Probability 38 (2010), 1947–1985.
R. O’Donnell, R. Servedio, L.-Y. Tan and A. Wan, A regularity lemma for low noisyinfluences, unpublished manuscript.
P. Raghavendra, Optimal algorithms and inapproximability results for every CSP?, in STOC’08—Proceedings of the 2008 ACM international Symposium on Theory of Computing, ACM, New York, 2008, pp. 245–254.
R. Rossignol, Noise-stability and central limit theorems for effective resistance of random electric networks, Annals of Probability 44 (2016), 1053–1106.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by NSF Grants CCF 1665252 and DMS-1737944 and DOD ONR grant N00014-17-1-2598.
Rights and permissions
About this article
Cite this article
Mossel, E. Gaussian bounds for noise correlation of resilient functions. Isr. J. Math. 235, 111–137 (2020). https://doi.org/10.1007/s11856-019-1951-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-019-1951-x