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Noise stability offunctions with lowinfluences:Invariance andoptimality Elchanan MosselRyan O’DonnellKrzysztof Oleszkiewicz2010 год

Noise stability of functions with low influences: Invariance and optimality
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Аннотация: In this paper we study functions with low influences on product probability spaces.These are functions f W 1 n !‫ޒ‬ that have EOEVar i OEf small compared to VarOEf for each i .The analysis of boolean functions f W f 1; 1g n !f 1; 1g with low influences has become a central problem in discrete Fourier analysis.It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics.We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces.Ours is one of the very few known nonlinear invariance principles.It has the advantage that its proof is simple and that its error bounds are explicit.We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly "smoothed"; this extension is essential for our applications to "noise stability"-type problems.In particular, as applications of the invariance principle we prove two conjectures: Khot, Kindler, Mossel, and O'Donnell's "Majority Is Stablest" conjecture from theoretical computer science, which was the original motivation for this work, and Kalai and Friedgut's "It Ain't Over Till It's Over" conjecture from social choice theory.
Год издания: 2010
Авторы: Elchanan Mossel, Ryan O’Donnell, Krzysztof Oleszkiewicz
Издательство: Princeton University
Источник: Annals of Mathematics
Ключевые слова: Complexity and Algorithms in Graphs, Game Theory and Voting Systems, Advanced Graph Theory Research
Другие ссылки: Annals of Mathematics (PDF)
Annals of Mathematics (HTML)
arXiv (Cornell University) (PDF)
arXiv (Cornell University) (HTML)
CiteSeer X (The Pennsylvania State University) (PDF)
CiteSeer X (The Pennsylvania State University) (HTML)