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On the Noise Sensitivity of Monotone Functions

  • Elchanan Mossel
  • Ryan O’Donnell
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

It is known that for all monotone functions f:{0, 1}n→ {0, 1}, if x∈ {0, 1}nis chosen uniformly at random and is obtained from by flipping each of the bits of independently with probability ∈, then \(P[{f_n}(x) \ne {f_n}(y)] < c \in \sqrt n\) for some c > 0. Previously, the best construction of monotone functions satisfying \(P[{f_n}(x) \ne {f_n}(y)] \geqslant \delta\) where 0< δ < 1/2, required ∈ ≥ 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 \(0 < \delta < 1/2, P[{f_n}(x) \ne {f_n}(y)] \geqslant \delta\) with:
  • ∈ = c(δ)n−α for any α < 1/2, using the recursive majority function with arity k = k(α);

  • \(\in = c(\delta ){n^{ - 1/2}}{\log ^t}n\) for \(t = {\log _2}\sqrt {\pi /2} = .3257 \ldots\) using an explicit recursive majority function with increasing arities; and

  • \(\in = c(\delta ){n^{ - 1/2}}\), non-constructively following a probabilistic CNF construction due to Talagrand.

Keywords

Boolean Function Monotone Function Balance Function Noise Sensitivity Noise Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Elchanan Mossel
    • 1
  • Ryan O’Donnell
    • 2
  1. 1.Hebrew Universit of Jerusalem and Microsoft ResearchHebrewIsrael
  2. 2.MIT Mathematics DepartmentUSA

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