Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy

Niv Levhari*, Alex Samorodnitsky*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Let Tϵ, 0≤ϵ≤1/2, be the noise operator acting on functions on the boolean cube {0,1}n. Let f be a distribution on {0,1}n and let q>1. We prove tight Mrs. Gerber-type results for the second Rényi entropy of Tϵf which take into account the value of the qth Rényi entropy of f. For a general function f on {0,1}n we prove tight hypercontractive inequalities for the ℓ2 norm of Tϵf which take into account the ratio between ℓq and ℓ1 norms of f.

Original languageAmerican English
Article number1376
JournalEntropy
Volume24
Issue number10
DOIs
StatePublished - 27 Sep 2022

Bibliographical note

Publisher Copyright:
© 2022 by the authors.

Keywords

  • Mrs. Gerber’s inequality
  • Rényi entropy
  • entropy
  • hypercontractivity

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