Invariance principle on the slice

Yuval Filmus, Guy Kindler, Elchanan Mossel, Karl Wimmer

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

The non-linear invariance principle of Mossel, O'Donnell and Oleszkiewicz establishes that if f(x1, . . . , xn) is a multilinear low-degree polynomial with low influences then the distribution of f(B1, . . . , Bn) is close (in various senses) to the distribution of f(G1, . . . , Gn), where Bi ϵR {-1, 1} are independent Bernoulli random variables and Gi ∼N(0, 1) are independent standard Gaussians. The invariance principle has seen many application in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAXCUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (X1, . . . ,Xn), (Y1, . . . ,Yn) such that (i) Matching moments: Xi and Yi have matching first and second moments, (ii) Independence: The variables X1, . . . ,Xn are independent, as are Y1, . . . ,Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions (X1, . . . ,Xn) in which the individual coordinates are not independent. A common example is the uniform distribution on the slice [n] k which consists of all vectors (x1, . . . , xn) ϵ {0, 1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdos-Ko-Rado theorems) and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1, . . . ,Xn) is the uniform distribution on a slice [n] pn and (Y1, . . . ,Yn) consists either of n independent Ber(p) random variables, or of n independent N(p, p(1 - p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain's tail theorem, a version of the Kindler- Safra structural theorem, and a stability version of the t-intersecting Erdos-Ko-Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.

Original languageAmerican English
Title of host publication31st Conference on Computational Complexity, CCC 2016
EditorsRan Raz
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages15:1-15:10
ISBN (Electronic)9783959770088
DOIs
StatePublished - 1 May 2016
Event31st Conference on Computational Complexity, CCC 2016 - Tokyo, Japan
Duration: 29 May 20161 Jun 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume50
ISSN (Print)1868-8969

Conference

Conference31st Conference on Computational Complexity, CCC 2016
Country/TerritoryJapan
CityTokyo
Period29/05/161/06/16

Bibliographical note

Publisher Copyright:
© Yuval Filmus, Guy Kindler, Elchanan Mossel, and Karl Wimmer.

Keywords

  • Analysis of boolean functions
  • Invariance principle
  • Johnson association scheme
  • The slice

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