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

doi:10.3390/e24101376

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

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

Let (Formula presented.), (Formula presented.), be the noise operator acting on functions on the boolean cube (Formula presented.). Let f be a distribution on (Formula presented.) and let (Formula presented.). We prove tight Mrs. Gerber-type results for the second Rényi entropy of (Formula presented.) which take into account the value of the (Formula presented.) Rényi entropy of f. For a general function f on (Formula presented.) we prove tight hypercontractive inequalities for the (Formula presented.) norm of (Formula presented.) which take into account the ratio between (Formula presented.) and (Formula presented.) norms of f.

Original language	American English
Article number	1376
Pages (from-to)	1-27
Number of pages	27
Journal	Entropy
Volume	24
Issue number	10
DOIs	https://doi.org/10.3390/e24101376
State	Published - Oct 2022

Bibliographical note

Publisher Copyright:
© 2022 by the authors.

Keywords

Mrs. Gerber’s inequality
Rényi entropy
entropy
hypercontractivity

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10.3390/e24101376

Cite this

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abstract = "Let (Formula presented.), (Formula presented.), be the noise operator acting on functions on the boolean cube (Formula presented.). Let f be a distribution on (Formula presented.) and let (Formula presented.). We prove tight Mrs. Gerber-type results for the second R{\'e}nyi entropy of (Formula presented.) which take into account the value of the (Formula presented.) R{\'e}nyi entropy of f. For a general function f on (Formula presented.) we prove tight hypercontractive inequalities for the (Formula presented.) norm of (Formula presented.) which take into account the ratio between (Formula presented.) and (Formula presented.) norms of f.",

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author = "Niv Levhari and Alex Samorodnitsky",

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AB - Let (Formula presented.), (Formula presented.), be the noise operator acting on functions on the boolean cube (Formula presented.). Let f be a distribution on (Formula presented.) and let (Formula presented.). We prove tight Mrs. Gerber-type results for the second Rényi entropy of (Formula presented.) which take into account the value of the (Formula presented.) Rényi entropy of f. For a general function f on (Formula presented.) we prove tight hypercontractive inequalities for the (Formula presented.) norm of (Formula presented.) which take into account the ratio between (Formula presented.) and (Formula presented.) norms of f.

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

Abstract

Bibliographical note

Keywords

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