Statistics Papers
Document Type
Journal Article
Date of this Version
102003
Publication Source
Random Structures & Algorithms
Volume
23
Issue
3
Start Page
333
Last Page
350
DOI
10.1002/rsa.10097
Abstract
It is known that for all monotone functions f : {0, 1}^{n} → {0, 1}, if x ∈ {0, 1}^{n} is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ϵ = n^{−α}, then P[f(x) ≠ f(y)] < cn^{−α+1/2}, for some c > 0.
Previously, the best construction of monotone functions satisfying P[f_{n}(x) ≠ f_{n}(y)] ≥ δ, where 0 < δ < 1/2, required ϵ ≥ c(δ)n^{−α}, where α = 1 − ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[f_{n}(x) ≠ f_{n}(y)] ≥ δ, with:

ϵ = c(δ)n^{−α} for any α < 1/2, using the recursive majority function with arity k = k(α);

ϵ = c(δ)n^{−1/2}log^{t}n for t = log_{2} = .3257 …, using an explicit recursive majority function with increasing arities; and

ϵ = c(δ)n^{−1/2}, nonconstructively, following a probabilistic CNF construction due to Talagrand.
We also study the problem of achieving the best dependence on δ in the case that the noise rate ϵ is at least a small constant; the results we obtain are tight to within logarithmic factors.
Copyright/Permission Statement
This is the peer reviewed version of the following article: Mossel, E. and O'Donnell, R. (2003), On the noise sensitivity of monotone functions. Random Struct. Alg., 23: 333–350., which has been published in final form at doi: 10.1002/rsa.10097. This article may be used for noncommercial purposes in accordance with Wiley Terms and Conditions for SelfArchiving http://olabout.wiley.com/WileyCDA/Section/id820227.html#terms.
Recommended Citation
Mossel, E., & O'Donnell, R. (2003). On the Noise Sensitivity of Monotone Functions. Random Structures & Algorithms, 23 (3), 333350. http://dx.doi.org/10.1002/rsa.10097
Date Posted: 27 November 2017
This document has been peer reviewed.