Statistics Papers

Document Type

Journal Article

Date of this Version

10-2003

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[fn(x) ≠ fn(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[fn(x) ≠ fn(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−1/2logtn for t = log2 = .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 non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving http://olabout.wiley.com/WileyCDA/Section/id-820227.html#terms.

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Date Posted: 27 November 2017

This document has been peer reviewed.