Concentration on the Boolean hypercube via pathwise stochastic analysis

Ronen Eldan, Renan Gross

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions.Second, we strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function f, (f)≤ C∑ i=1 n i (f)/1+log(1/ i (f)). We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn-Kalai-Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary.Lastly, we improve a quantitative relation between influences and noise stability given by Keller and Kindler.Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs.
Original languageEnglish
Title of host publicationSTOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on theory of computing
EditorsKonstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy
Pages208-221
Number of pages14
ISBN (Electronic)9781450369794
DOIs
StatePublished - 8 Jun 2020
EventSTOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing - Chicago, IL, USA
Duration: 22 Jun 202026 Jun 2020

Publication series

NameAnnual ACM SIGACT Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

ConferenceSTOC 2020: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
Period22/06/2026/06/20

Keywords

  • Boolean analysis
  • Concentration
  • Isoperimetric inequality
  • Pathwise analysis

All Science Journal Classification (ASJC) codes

  • Software

Fingerprint

Dive into the research topics of 'Concentration on the Boolean hypercube via pathwise stochastic analysis'. Together they form a unique fingerprint.

Cite this